This is a preprint of: “Three-dimensional Lotka-Volterra systems with 3:-1:2-resonance”, Waleed Aziz, Colin J. Christopher, Jaume Llibre, Chara Pantazi, Mediterranean J. Math., vol. 18, 167:1– 24, 2021. DOI: [10.1007/s00009-021-01809-2] Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance Waleed Aziz1 , Colin Christopher2 , Jaume Llibre3 and Chara Pantazi4 Abstract. We study the local integrability at the origin of a nine para- metric family of a three dimensional Lotka–Volterra differential systems with 3:-1:2-resonance. We present the necessary and sufficient conditions on the parameters of the family that guarantee the existence of two in- dependent local first integrals at the origin of coordinates. Additionally we check the cases where the origin is linearizable. Mathematics Subject Classification (2010). Primary 34A05, 34C20; Sec- ondary 34A34, 34C14. Keywords. Lotka Volterra; Integrability; Linearizability; Jacobi multi- plier. 1. Introduction and statement of the main results Lotka–Volterra systems [30, 46] have been used in population biology to de- scribe the evolution of conflicting species, see for example [37, 38]. During the years these systems have been used in many branches of sciences like laser physics [27], neural networks [40], plasma physics [28], chemical kinetics [39], etc. The dynamical behaviour of these models has been widely studied, see for instance [1, 14, 8, 45, 49, 34] among others. The integrability of some Lotka–Voltera families was considered by several authors, see for example [15, 16, 18, 31, 34, 36]. Similar technics applied to some more general families like the ones in [5, 23]. Some general properties of these systems are studied in the works [12, 13]. In general, for a three dimensional system, the existence of a first in- tegral is very helpful because reduces the study of the dynamics from three dimensions to two. Moreover, the knowledge of two independent first integrals determines completely the trajectories of the system. Hence, the study of the existence of first integrals is an important issue in the qualitative theory of dynamical systems. There are many technics for the search of first integrals 2 W. Aziz, C. Christopher, J. Llibre and C. Pantazi of a differential system. Some of them are: Lax pairs [3, 29], Lie symmetries [2, 41], Noether symmetries [17], Darboux theory of integrability [21, 22], Painlevé analysis [9], Differential Galois Theory [44, 48]. Here we use the same method which was employed in [4, 6] for studying the local integrability at the origin of all the Lotka–Voltera three dimensional differential systems of the form ẋ = x(3 + ax + by + cz) = P (x, y, z), ẏ = y(−1 + dx + ey + f z) = Q(x, y, z), (1) ż = z(2 + gx + hy + kz) = R(x, y, z). We consider the case of (3:-1:2)-resonance in order to continue the work in [6]. Note that we are working in the Siegel domain with two independent resonances. Moreover, when restricted to y=0, the nodal behaviour has no resonance. Hence, using the behaviour associated to the resonance of the node we tried to find the generic behaviour of the integrability of these systems. It seems that the problem still is very complicated and more cases should be studied in order to find out the general mechanisms for obtaining the inte- grability. Our aim is to understand the integrability mechanisms for general resonances. The main result of this work is Theorem 1 where we characterize all the Lotka–Voltera differential systems (1) having two independent local first integrals. We provide a complete classification for the integrability and lin- earizability conditions of systems (1) with (3 : −1 : 2)-resonance. Theorem 1. Consider the three dimensional Lotka-Volterra system with (3 : −1 : 2)-resonance. The origin of system (1) is integrable if and only if one of the following conditions hold. 1) r1 = 2ef − ek − hk = 0, r2 = 2cd + cg − 5dk + 3f g − gk = 0 r3 = 2bk − 2ce + 3ek − 3hk = 0, r4 = 2bf − bk − ch − 3f h + 3hk = 0 r5 = 2bd + bg − 2de − 3dh + 2eg = 0, r6 = 2af + ak − 2dk − gk = 0 r7 = 2ae + ah − 2de − eg = 0, r8 = 2ac − 5ak − 2cd − cg + 5dk − 3f g + 4gk = 0 r9 = 2ab − 2ah − 2bd − bg + 3dh = 0 2) b + 2e = c = d = f = k = 0 3) a = e + h = f = d = g = 0 4) a = d = g = h = 0 5) b = c = f = k = 0 6) a − d = b − e = g = h = 0 7) b = c = e − h = f − k = 0 Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 3 8) b = h = 0 9) b + e = c = f = k = 0 10) h = b + e = 0 11) b + 2e = d = h = 0 12) 8a − 3g = b + 3e = 8d + g = h = 0 13) b = e + h = f = 0 14) b − h = e + h = f = 0 15) b − 2h = d = e + h = f = 0 16) a − 2d − g = e + h = f = b + h = d + g = 0 17) b = 2c − 9k = 2e + h = 2f + k = 0 18) 2b − h = 2c − 7k = 2e + h = 2f + k = 0 19) b + h = c + k = e − h = f − k = 0 20) b + 2h = c + 2k = d = e − h = f − k = 0. Additionally, system (1) is linearizable if and only if either one of the con- ditions 2 − 11, 13 − 16, 19, 20 or one of the following subcases of case 1) holds. 1.1) dh − eg = k = f = c = b − e = a − d 1.2) k = g = f = d = c = a 1.3) f − k = e − h = d − g = c − k = b − h = a − g = 0 1.4) bk − ch = g = f − k = e − h = d = a. The basic notions that we use for the proof of Theorem 1 are presented in Section 2. Then, in Section 3 we give the proof of Theorem 1. 2. Preliminaries Consider the three dimensional system (1) and the corresponding vector field X given by ∂ ∂ ∂ X =P +Q +R . ∂x ∂y ∂z Let U be an open neighborhood of the origin of C3 . We say that a non-locally constant analytic function H : U → C is a local first integral of system (1) if it is constant on all solutions (trajectories) of the system contained in U. Hence, the first integral H of system (1) satisfies ∂H ∂H ∂H XH = P +Q +R = 0, ∂x ∂y ∂z in all the points of U. Two local first integrals H1 and H2 defined in U are independent if their gradients are linearly independent in C3 (except perhaps in a set of Lebesgue measure zero). 4 W. Aziz, C. Christopher, J. Llibre and C. Pantazi An inverse Jacobi multiplier of X is a function M : U → C that satisfies the equation X (M ) = M divX in U, or, equivalently, div(X /M ) = 0. Here divX = ∂P/∂x + ∂Q/∂y + ∂R/∂z. For general properties of the Darboux Jacobi multipliers see [7] and references therein. One method for finding two local independent first integrals consists in finding an inverse Jacobi multiplier and a local first integral because then we can use the Jacobi multiplier to obtain the second first integral, see [26]. Another way of finding two local independent first integrals is to show that the system (1) is integrable, following the works of [10, 11, 43, 47]. That is, there is an analytic change of coordinates near to identity  (X, Y, Z) = x + o(x, y, z), y + o(x, y, z), z + o(x, y, z) , (2) which transforms system (1) into the system Ẋ = 3Xζ(x, y, z), Ẏ = −Y ζ(x, y, z), Ż = 2Zζ(x, y, z), (3) where ζ = 1 + o(x, y, z). Note that XY 3 and Y 2 Z are first integrals of system (3) and we can pull them back to the first integrals of system (1) of the form φ1 = xy 3 (1 + O(x, y, z)), φ2 = y 2 z(1 + O(x, y, z)). (4) Conversely, given two independent first integrals φ1 and φ2 of system (1) of the form (4), then there exits a change of coordinates X, Y, Z so that these first integrals can be written in the form XY 3 and Y 2 Z and for some ζ the new system is of the form (3). We say that system (1) is linearizable at the origin if and only if the change of coordinates (2) can be chosen such that ζ ≡ 1. Hence, system (1) can be rewritten as Ẋ = 3X, Ẏ = −Y, Ż = 2Z. Hence, we can choose the transformation (2) such that the coefficients of the non–linear terms in ζ(x, y, z) vanish, see also [19, 20, 25]. Let ` = `(x, y) be a polynomial. An invariant algebraic surface ` = 0 of the polynomial differential system (1) is an algebraic surface which satisfies ∂` ∂` ∂` `˙ = X ` = P +Q +R = `L` , (5) ∂x ∂y ∂z for some polynomial L` ∈ C[x, y, z]. The polynomial L is called a cofactor of the invariant algebraic surface ` = 0. Note that from relation (5) we have that any cofactor has at most degree one because the polynomial vector field has degree two. There are other functions which satisfy (5), for example there are invariant analytic surfaces. Then, the degree of the cofactor L` is imposed rather than following from equation (5). Exponential factors also satisfy (5) and are related to the multiplicity of the invariant surfaces, see [32, 33]. Exponential factors appear for those Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 5 parameter values where two surfaces collapse or an invariant surface is coin- cident with the plane at infinity. Let E(x, y, z) = exp(f (x, y, z)/g(x, y, z)), where f, g ∈ C[x, y, z] are relatively prime polynomials. We say that E in an exponential factor of (1) if X E = ELE , (6) for some polynomial LE of degree at most one, and LE is called the cofactor of the exponential factor E. Invariant algebraic surfaces and exponential factors have an important role in the explicit calculation of first integrals. More concretely, if there is a linear combination between their cofactors, then the existence of a first integral is immediate, as the following theorem shows, [21, 22]. Theorem 2. Let `1 , · · · , `p invariant algebraic surfaces of system (1) with co- factors L`1 , · · · , L`p and let E1 , · · · , Eq be exponential factors of system (1) with cofactors LE1 , · · · , LEq . If there exist real numbers α1 , · · · , αp , β1 , · · · , βq , not all of them equal to zero, such that α1 L`1 + · · · + αp L`p + β1 E1 + · · · + βq Eq = 0, then system (1) has a first integral (called a Darboux first integral) αp β1 H = `α βq 1 · · · `p E1 · · · Eq . 1 The following two theorems are crucial in our study and they are proved in [6]. We use the multi–index notation X I = xi y j z k in order to simplify the presentation of Theorem 3. Theorem 3 (Theorem 1 of [6]). Suppose the analytic vector field       X ∂ X ∂ X ∂ x λ + AxI X I  +y µ + AyI X I  +z ν + AzI X I  , ∂x ∂y ∂z |I|>0 |I|>0 |I|>0 has a first integral φ = xα y β z γ (1 + O(x, y, z)) with at least one of α, β, γ 6= 0 and an inverse Jacobi multiplier M = xr y s z t (1 + O(x, y, z)) and suppose that the cross product of (r − i − 1, s − j − 1, t − k − 1) and (α, β, γ) is bounded away from zero for any integers i, j, k ≥ 0. Then the vector field has a second analytic first integral of the form ψ = x1−r y 1−s z 1−t (1 + O(x, y, z)) and hence system (1) is integrable. Remark 1. Theorem 3 can be generalized to allow the cross product to be either zero or bounded away from zero. In this case, there are resonant terms which must be shown to be zero when the cross product is zero. We do not need this more detailed case here as alternative proofs are given in cases where the cross product can vanish. The proof of the theorem follows directly from applying a change of coordinates, and a scaling, so that the quantities φ and M are just simple monomials, and then working term by term. 6 W. Aziz, C. Christopher, J. Llibre and C. Pantazi Remark 2. It can happen that the second first integral is also of the form ψ = (xα y β z γ )λ (1 + O(x, y, z)) for some λ. However, it follows from the proof of Theorem 3, that this second integral will be functionally independent of φ. Therefore, ψ 1/λ /φ − 1 is an analytic first integral which must also be expressible in the form xδ y  z ζ (1 + O(x, y, z)) for some δ, , ζ. If (δ, , ζ) is a multiple of (α, β, γ) then we can repeat this process, eventually ending up with a second first integral of the form above with (δ, , ζ) independent of (α, β, γ), as required to give two first integrals in the form (4). Hence for system (1), in normal situation, from the expression of a first integral and an inverse Jacobi multiplier we can obtain the explicit expression of the other first integral. Theorem 4 (Theorem 2 of [6]). Suppose that we have a three dimensional Lotka-Volterra system with (λ : µ : ν)-resonance, for which the origin is integrable, and there exists a function ξ = xα y β z γ (1 + O(x, y, z)) such that X(ξ) = kξ for some constant k = αλ + βµ + γν 6= 0, then the system is linearizable. Proof. From the hypothesis, we can bring the system to the form ẋ = λxζ, ẏ = µyζ, ż = νzζ. Let ξ = xα y β z γ g(x, y, z), then the linearizing change of coordinates is given by (X, Y, Z) = (x g λ/k , y g µ/k , z g ν/k ). (7)  An ordered tuple of the complex numbers (λ1 , λ2 , λ3 ) in C3 is called resonant if there exist non negative integers α = (α1 , α2 , α3 ) ∈ Z3+ such that α1 λ1 + α2 λ2 + α3 λ3 = λj , |α| = α1 + α2 + α3 ≥ 2. (8) The natural number |α| = α1 + α2 + α3 is the order of the resonance and the monomial X n = xα1 y α2 z α3 ej is called a resonant monomial. The coefficient of the monomial X n in the system (1) is called a resonant coefficient and the corresponding term is called a resonant term. The Poincaré domain is the collection of all tuples (λ1 , λ2 , λ3 ) such that the convex hull of the point set {λ1 , λ2 , λ3 } ∈ C does not contain the origin inside or on the boundary. It is well known that singular points whose eigenvalues lie in the Poicaré domain can be brought, via an analytic change of coordinates, to a normal form which contains only resonant terms. Then, there are only a finite number of resonant terms that will need to be checked in order to show linearizability. 3. Proof of Theorem 1 After some calculations we see that systems (1) admit the invariant hyper- planes x = 0, y = 0, z = 0 with cofactors Lx = 3 + ax + by + cz, Ly = −1 + dx + ey + f z, Lz = 2 + gx + hy + kz respectively. Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 7 Our aim is to find conditions that guarantees the existence of two inde- pendent analytic first integrals of system (1) of the form φ1 = xy 3 (1 + O(x, y, z)) and φ2 = y 2 z(1 + O(x, y, z)). (9) So, we first express φ1 and φ2 as power series up to terms of order 18. Then we compute the obstructions that φ1 and φ2 form first integrals. A factorized Gröbner basis is obtained, giving the necessary conditions for integrability and linearizability described in Theorem 1. To achieve this, we have used both Maple and Reduce. Also we have use the minAssGTZ algorithm of Singular [42] in order to check that the conditions found were irreducible. The reader can find more details about the mechanism that we have applied in Section 3 of [6]. Proof of Theorem 1. Here, in order to complete the proof of Theorem 1 we shall show below that each of these conditions is also sufficient for the inte- grability of system (1). Case 1 If k 6= 0 the system has the invariant plane ` = 1 + a3 x − ey + k2 z = 0 with cofactor L` = ax + ey + kz. Indeed, the relation (5) gives        ab ae ac gk ak hk ek − de + 2 y+ + −5 z x+ −ef + + zy = 0. 3 3 3 2 6 2 2 If we set ba ea ca gk ka hk ke c1 = − de + 2 , c2 = + −5 , c3 = −f e + + , 3 3 3 2 6 2 2 and note that c1 = r5 /6 + r7 /3 + r9 /6, c2 = r2 /6 + r8 /6, c3 = −r1 /2, then c1 = c2 = c3 = 0 is a direct consequence of the conditions satisfied for this case. Now, using the Darboux theory of integrability (see Theorem 2 and for more details chapter 8 of [24]), we obtain the two independent first integrals 3f +c 2f φ1 = xy 3 `− k , φ2 = y 2 z`1− k . If k = 0 we consider the following two subcases. Subcase 1 k = a = e = 2bd + bg − 3dh = 2bf − ch − 3f h = 2cd + cg + 3f g = 0. i) b 6= 0 and h 6= 0. We get the exponential factor E = exp(dhx − bhy + bf z) with cofactor LE = 3dhx + bhy + 2bf z. Thus, by the Darboux theory of integrability, system (1) admits the two first integrals 1 1 φ1 = xy 3 `− h , φ2 = y 2 z`− b . ii) h = 0 and b 6= 0. We distinguish the following subcases. a) a = 2d + g = e = f = h = k = 0. System (1) admits the exponential factor E = exp (dx − by + cz/2) with cofactor LE = 3d + by + cz. In this case we obtain the two independent analytic first integrals φ1 = xy 3 E −1 , φ2 = y 2 z. 8 W. Aziz, C. Christopher, J. Llibre and C. Pantazi b) 2a − 2d − g = c = f = h = k = 2bd + bg − 2de + 2eg. If e 6= 0 we obtain the invariant hyperplane ` = 1 + a3 x − ey with cofactor L` = ax + ey and the two independent analytic first integrals b φ1 = x y 3 `−3− e , φ2 = y 2 z`−2 . If e = 0 we have an exponential factor E = exp(dx − by) with cofactor LE = 3dx + by. System (1) admits the two analytic inde- pendent first integrals φ1 = x y 3 E −1 , φ2 = y 2 z. (iii) a = b = d = e = k = c + 3f = 0. There is an exponential factor E = exp(gx/3 − hy + f z) with cofactor LE = gx + hy + 2f z and we obtain the two analytic first integrals φ1 = xy 3 E, φ2 = y 2 zE −1 . iv) If b = h = 0 then we fall in Case 8. Subcase 2 k = c = f = 2ab − 2ae − 3ah + 3eg = 2ae + ah − 2de − eg = 2bd + bg − 2de − 3dh + 2eg = 0. When e 6= 0 the system has the invariant plane ` = 1 + a3 x − ey = 0 with cofactor L` = ax + ey and we obtain the two independent first integrals b h φ1 = x y 3 `−3− e , φ2 = y 2 z `−2+ e . When e = 0 the system (1) admits the exponential factor E = exp(dx − by) with cofactor LE = 3dx + by and then we can construct the two independent first integrals h φ1 = x y 3 E −1 , φ2 = y 2 z E − b , if b 6= 0. When b = 0 we can reduce to Case 5. Case 1.1 For e 6= 0 the system has the invariant plane ` = 1 + d3 x − ey = 0 with cofactor L` = dx + ey and a linearizing change of coordinates is given by   h (X, Y, Z) = x`−1 , y`−1 , z`− e . For e = 0 we distinguish two subcases. First, for h = 0 and d 6= 0 we get a linearizing change of coordinates  g  (X, Y, Z) = x`−1 , y`−1 , z`− d . If d = 0, a linearizing change of coordinates is given by   g  (X, Y, Z) = x, y, z exp hy − x . 3 Case 1.2 When e 6= 0, the linearizing change of coordinates is  b h  (X, Y, Z) = x(1 − ey)− e , y(1 − ey)−1 , z(1 − ey)− e . Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 9 When e = 0, the linearizing change of coordinates is (X, Y, Z) = (x exp(by), y, z exp(hy)). Case 1.3 In this case the linearizing change of coordinates is  (X, Y, Z) = x`−1 , y`−1 , z`−1 , where ` = 1 + a3 x − by + 2c z. Case 1.4 When h 6= 0, the linearizing change of coordinates is given by  b  (X, Y, Z) = x`− h , y`−1 , z`−1 , where ` = 1 − hy + k2 z. When h = 0 we have two possibilities. First for b = 0 and k 6= 0 then we have a linearizing change of coordinates c  (X, Y, Z) = x`− k , y`−1 , z`−1 . Second, if k = 0, then the linearizing change of coordinates is   c   (X, Y, Z) = x exp by − z , y, z . 2 In order to facilitate the understanding of the proof we group the remaining cases according to the techniques used to prove integrability. Group 1. In Cases 2 and 3 the variables x and z do not appear in the differ- ential equation respect to the variable y. Moreover, we can obtain a sufficient number of invariant algebraic surfaces and exponential factors to construct two first integrals via Theorem 2. Case 2 The system becomes ẋ = x(3 + ax − 2ey), ẏ = y(−1 + ey), ż = z(2 + gx + hy). For ae 6= 0 this system has the invariant plane `1 = 1 − ey = 0 and the invariant algebraic surface `2 = 1 + (a/3)x + (ae/3)xy + (ae2 /3)xy 2 = 0 with cofactors L`1 = ey and L`2 = ax, respectively. Here we obtain the two independent first integrals g − 13 −(2+ h e ) −a φ1 = x y 3 `21 `2 and φ2 = y 2 z `1 `2 . When e = 0 and a 6= 0 the system has the invariant plane ` = 1 + ax/3 and the exponential factor E = exp(y) with cofactors L` = ax and LE = −y which yield to the two independent first integrals g φ1 = x y 3 `−1 and φ2 = y 2 z `− a E h . While a = 0 and e 6= 0 we have an invariant plane ` = 1 − ey and an exponential factor E = exp(x/(1 − ey)) with cofactors L` = ey and LE = 3x and we obtain the two independent first integrals h g φ1 = x y 3 `−1 and φ2 = y 2 z `−2− e E − 3 . 10 W. Aziz, C. Christopher, J. Llibre and C. Pantazi Finally when a = e = 0 there exist exponential factors E1 = exp(x) and E2 = exp(y) with cofactors LE1 = 3x and LE2 = −y and we have the two independent first integrals − g3 φ1 = x y 3 and φ2 = y 2 z E1 E2h . Theorem 4 guarantees the linearizability of the system in this case as ξ = y/(1 − ay/3) satisfies ξ˙ = −ξ. Case 3 The system takes the form ẋ = x(3 + by + cz), ẏ = y(−1 − hy), ż = z(2 + hy + kz). The system has the invariant plane `1 = 1+hy = 0 and the invariant algebraic surface `2 = 1+kz/2−(kh/2)yz = 0 with cofactors L`1 = −hy and L`2 = kz. When h 6= 0 and k 6= 0 then we have a linearizing change of coordiates  b c  − (X, Y, Z) = x`1h `2 k , y`−1 1 , z`1 `−1 2 . b When h = 0 we replace `1h by the exponential factor exp(by), and when k = 0 −c we replace `2 k by exp(cz(1 − hz)/2). Group 2. Similarly to the previous group, in Cases 4 and 5 there is also a differential equation which involves only one variable. But in this group we only obtain one invariant algebraic surface from which we are able to construct a first integral and an inverse Jacobi Multiplier. From Theorem 3 we can construct another first integral. Additionally, a linearizing transformation can be constructed. Case 4 We provide two different proofs. In the first we prove the existence of two independent local first integrals. In the second we provide, for generic parameter values, one first integral and an Inverse Jacobi Multiplier. The system is ẋ = x(3 + by + cz), ẏ = y(−1 + ey + f z), ż = z(2 + kz). The system for k 6= 0 has an algebraic invariant surface ` = 1 + k2 z = 0 with cofactor L` = kz. Using the change of coordinates Z = z`−1 we can linearize the last equation and bring it into the form Ż = 2Z. To linearize the second equation we seek an invariant surface of the form `˜ = A + By = 0 with A = A(Z) and A(0) = 1 and B = B(Z) with cofactor L`˜ = ey + f z. To find A and B we need to solve the system of the two differential equations Ȧ = Af z, Ḃ − B = Ae. For k = 0 we have A = exp (f Z/2) whereas for k 6= 0 we have A = −f /k (1 − kZ/2) . P P Now write A = i≥0 ai Z i and B = i≥0 bi Z i . From the relation Ḃ − B = Ae we have bi = e ai /(2i − 1) and so B is a convergent series. Hence the second equation is linearized and taking Y = y/`˜ we get Ẏ = −Y . In order Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 11 to linearize the first equation we set X = x exp (−γ(Y, Z)) with γ an analytic function satisfying γ̇ = b y(Y, Z) + c z(Z). We set X X X γ(Y, Z) = γij Y i Z j , y(Y, Z) = ηij Y i Z j , z(Z) = ζj Z j , i+j>0 i+j>0 j>0 and hence, we need to solve X X X (2j − i)γij Y i Z j = b ηij Y i Z j + c ζj Z j . i+j>0 i+j>0 j>0 If y(Y, Z) contains no terms of the form (Y 2 Z)n we obtain b η0j + c ζj b ηij γ0j = , γij = , i > 0. 2j 2j − i Clearly γ is a convergent series. Now we should show that the inverse trans- formation y = AY /(1 − BY ) contains no term like (Y 2 Z)n . Note that AY X y= = A B k Y k+1 . 1 − BY We assume that k + 1 = 2n for some n. We should show that AB k = AB 2n−1 contains no term like Z n . Note that dB dB Ḃ = = 2Z . dt dZ From equation Ḃ − B = Ae we obtain that  2Z dB 2n  2n −B = e A B 2n−1 . 2n dZ Now on the left hand side the coefficient in Z n in B 2n vanishes. So either e = 0, that means B ≡ 0, or the coefficient of Z n in AB 2n−1 vanishes. Therefore y given by Y = y/(A + By) contains no term like (Y 2 Z)n . Thus we have set up the existence of a linearizing transformation and this completes the first proof. Now we will provide the second proof. When bk 6= 0 the system has the first integral  2ce−3ek−2be−bk −2ek 2bk (b+3e)k 1 φ=x y z 1 + kz 2 , and the inverse Jacobi multiplier 2e 1 3e 1 3 f 3e 2ce IJM = x1+ b z 2 − b (1 + kz) 2 + k + b − bk . 2 Theorem 3 guarantees the existence of the second first integral 2e 1 3e ψ = x− b yz 2 + b (1 + . . .). The desired first integrals of the initial system are 1 b+6e b 2 φ1 = φ 2 bke ψ −3− e = xy 3 (1 + . . .) and φ2 = φ bk ψ −2 = y 2 z(1 + . . .). For the remaining cases, we can work in a similar way. Since ξ = z`−1 satisfies ξ˙ = 2ξ, the system is linearizable by Theorem 4. 12 W. Aziz, C. Christopher, J. Llibre and C. Pantazi Case 5 The system becomes ẋ = x(3 + ax), ẏ = y(−1 + dx + ey), ż = z(2 + gx + hy). We will provide two different proofs as we did in Case 4. In the first one the system has the invariant plane ` = 1 + (a/3)x = 0 with cofactor L` = ax. The change of coordinates X = x `−1 will linearize the first equation and taking Y = y/(A(X) + B(X)y) will then linearize the second equation provided `˜ = A+By = 0 is an invariant surface with cofactor L`˜ = d x + e y. Therefore it suffices to find A(X) and B(X) satisfying the equations Ȧ = d A x, Ḃ − B = e A, A(0) = 1. (10) d From the first equation of (10) weP can find A = (1 + ax/3) P a (when a = 0 we take A = exp (dx/3)). Write A = i≥0 ai X i and B = i≥0 bi X i . Clearly B satisfies the second equation if we set bi = eai /(3i − 1), and the covergence of B follows immediately from the convergence of A. The third equation would be linearizable by the transformation Z = z exp(−γ(X, Y )) where γ̇ = g x(X) + h y(X, Y ). To find such γ we write X X X γ(X, Y ) = γij X i Y j , x(X) = ξi X i , y(X, Y ) = ηij X i Y j . i+j>0 i>0 i+j>0 It is easy to see that from relation γ̇ = g x(X) + h y(X, Y ) we obtain X X X (3i − j)γij X i Y j = g ξi X i + h ηij X i Y j . If y(X, Y ) contains no terms of the form (XY 3 )k , we can set g ξi + h ηij γij = , 3i − j to give a convergent expansion of γ. Now it remains to show that the inverse transformation y = AY /(1 − BY ) contains no term like (XY 3 )k . Note that AY X y= = A B k Y k+1 . 1 − BY Assume that k + 1 = 3n for some n. We need to show that AB k = AB 3n−1 contains no term like X n . Note that dB dB Ḃ = = 3X . dt dX From Equation (10) we obtain that  3X dB 3n  e A B 3n−1 = − B 3n . (11) 3n dX Clearly, the coefficient in X n in B 3n vanishes on the right hand side so that either e = 0, that means B ≡ 0, or the coefficient of X n in AB 3n−1 vanishes. Therefore y in Y = y/(A + By) contains no term like (XY 3 )k and we have a linearizing transformation. This completes the first proof. Now in the second proof we will just give the form of the first integrals and the inverse Jacobi multiplier since this procedure is exactly the same as Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 13 Case 4. We do not tackle the cases where the cross product is zero since we have already given a complete proof above. In cases (iii) and (iv), the integral obtained from Theorem 3 is of the same form and so we must rely on the arguments given in Remark 2 after Theorem 3. i) When ae 6= 0 we have  2ae−3eg+ah+3dh 1 φ = x−a(2e+h) y −3ah z 3ae 1 + ax 3 ,   3eg−3ed−ah−3dh ) h 2+ 3e 2+h/e 1 3ae IJM = x y 1 + ax 3 , h h ψ = x−1− 3e y −1− e z(1 + . . .), 1 φ1 = φ ae ψ −3 = xy 3 (1 + . . .), 3e+h h φ2 = φ 3ae2 ψ −2− e = y 2 z(1 + . . .). ii) When a = 0 and e 6= 0 we obtain φ = x−2e−h y −3h z 3e exp ((dh − eg)x) ,   h ge − de − dh IJM = x2+ 3e y 2+h/e exp x , 3e h h ψ = x−1− 3e y −1− e z(1 + . . .), 1 φ1 = φ e ψ −3 = xy 3 (1 + . . .), 3e+h h φ2 = φ 3e2 ψ −2− e = y 2 z(1 + . . .). iii) When e = 0 and a 6= 0 we get a 3d 2 a 4 d φ = x y 3 (1 + x)−1− a , IJM = x 3 z(1 + x) 3 + a . 3 3 iv) When a = e = 0 we have   3 2 d φ = x y exp(−dx), IJM = x z exp 3 x . 3 Finally, since ξ = x`−1 satisfies ξ˙ = 3ξ, the system is linearizable according to Theorem 4. Group 3. This group is formed by Cases 6 and 7. Here, we also have a dif- ferential equation which involves only one variable and we can obtain one invariant algebraic surface. Thus we can construct one first integral and one inverse Jacobi Multiplier. Then by Theorem 3 we obtain the other first inte- gral. In this group we use Theorem 4 to obtain a linearizing transformation. Case 6 The system in this case is of the form ẋ = x(3 + ax + by + cz), ẏ = y(−1 + ax + by + f z), ż = z(2 + kz). 14 W. Aziz, C. Christopher, J. Llibre and C. Pantazi For k 6= 0 the system has the invariant plane ` = 1 + k2 z = 0 with cofactor L` = kz. In this case we obtain the fist integral c−f −2k φ = x−1 y z 2 ` k , and the inverse Jacobi multiplier −5 9k+2f +4c IJM = x3 z 2 ` 2k . For k = 0 the system admits the exponential factor E = exp(z/2) with co- factor LE = z. In this case we obtain the first integral φ = x−1 y z 2 E c−f and 5 the inverse Jacobi multiplier IJM = x3 z − 2 E f −2c . In both cases, Theorem 7 3 guarantees the existence of the second first integral ψ = x−2 y z 2 (1 + o(1)). Thus we can construct two independent first integrals of the desired form φ1 = φ7 ψ −4 = xy 3 (1 + o(1)) and φ2 = φ4 ψ −2 = y 2 z(1 + o(1)). Since ξ = z`−1 satisfies ξ˙ = 2ξ then by Theorem 4 the system is linearizable. Case 7 We have the system ẋ = x(3 + ax), ẏ = y(−1 + dx + ey + f z), ż = z(2 + gx + ey + f z). For a 6= 0 the system has the invariant plane ` = 1 + ax/3 = 0 with cofactor L` = ax. Additionally, the system has the first integral g−d−a φ = x y z −1 ` a , and the inverse Jacobi multiplier 7 3g−a−6d IJM = x 3 y 3 ` 3a . For a = 0 we obtain the first integral g−d φ = x y z −1 E 3 , and the inverse Jacobi multiplier 7 g−2d IJM = x 3 y 3 ` 3 , where E = exp(x). Then, by Theorem 3, there is a second first integral of the 4 form ψ = x− 3 y −2 z(1 + · · · ) and hence we obtain two independent analytic first integrals of the desired form −3 φ1 = (φψ) = xy 3 (1 + · · · ), φ2 = φ−4 ψ −3 = yz 2 (1 + · · · ). From Theorem 4, since ξ = x`−1 satisfies ξ˙ = 3ξ, we have that the system is linearizable. Group 4. Cases 8 and 9 both have two equations which involve only two vari- ables and so there is a linearizable node. The third equation can be linearized via a power series argument. Case 8 The system can be written into the form ẋ = x(3 + ax + cz), ẏ = y(−1 + dx + ey + f z), ż = z(2 + gx + kz). Note that the first and third equation yields a linearizable node. Hence, there exists an analytic transformation of the form X = x(1 + O(x, z)) and Z = Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 15 z(1 + O(x, z)) such that the two equations can be written as Ẋ = 3X and Ż = 2Z. Now in order to linearize the second equation we seek an invariant surface of the form ` = A + By = 0 with cofactor L` = dx + ey + f z and B = B(X, Z) and A = A(X, Z) such that A(0, 0) = 1. The change of variable Y = y/(A + By) will then linearize the second equation. In order to find such A and B we have to solve the following system of differential equations Ḃ − B = e A, Ȧ = (d x + f z)A. (12) We write A = exp (α(X, Z)) and from the second equationP of (12) we obtain α̇(X, Z) = d x(X, Z) + f z(X, Z). Suppose that α = i+j>0 αij X i Z j . Then X X α̇(X, Z) = (3i + 2j)αij X i Z j = d x(X, Z) + f z(X, Z) =: cij X i Z j . i+j>0 i+j>0 cij P i j Thus αij = 3i+2j for i + j > 0 and the convergence of i+j>0 cij X Z guarantees P the convergence of P α and hence of A. Now, we write A = 1 + i j i j i+j>0 aij X Z and B = −e+ P i+j>0 bij X Z . Then from the first equation eaij of (12) we find that B = e+ i+j>0 3i+2j−1 X i Z j which is clearly convergent. Hence, the system is linearizable. Case 9 The system is ẋ = x(3 + ax + by), ẏ = y(−1 + dx − by), ż = z(2 + gx + hy). The change of variables (X, Y ) = (x, xy) transforms the first two equations into Ẋ = X(3 + aX) + bY, Ẏ = Y (2 + (a + d)X), which is a linearizable node and hence there is a linearizing change of coor- dinates of the form (X̃, Ỹ ) = (X + bY + o(X, Y ), Y (1 + o(1))) such that X̃˙ = 3X̃ and Ỹ˙ = 2Ỹ . To linearize the third equation, assume that we have a function α = α(X̃, Ỹ ) such that α̇ = g x(X̃, Ỹ ) + h y(X̃, Ỹ ). Then the change of variable Z̃ = z exp(−α) will give Z̃˙ = 2 Z̃. Writing X g x(X̃, Ỹ ) + h y(X̃, Ỹ ) = aij X̃ i Ỹ j , P aij i j it is easy to see that α(X̃, Ỹ ) = 3i+2j X̃ Ỹ . The convergence of α is then clear. Thus, the first integrals of the linear system X̃ −2 Ỹ 3 and X̃ −2 Ỹ 2 Z̃ can be pulled back to get φ1 = x y 3 (1 + . . .) and φ2 = y 2 z(1 + . . .), and linearizability follows from Theorem 4, since X̃ = x(1 + . . .) satisfies X̃˙ = 3X̃. Group 5. The rest of the cases belong to this group. Here, the three differential equations involve all the variables. Each case is treated in a different manner. 16 W. Aziz, C. Christopher, J. Llibre and C. Pantazi Case 10 The system can be written as ẋ = x(3 + ax + by + cz), ẏ = y(−1 + dx − by + f z), ż = z(2 + gx + kz). The change of variable Y = x y transforms the system into the form ẋ = x(3+ax+cz)+bY, Ẏ = Y (2+(a+d)x+(c+f )z), ż = z(2+gx+kz). (13) The singular point at the origin of system (13) is then in the Poincaré domain and it is easy to checked that there are no resonances. Therefore there exists a change of coordinates of the form   X, Ỹ , Z = (x + bY + o(x, Y, z), Y (1 + o(1)) , z (1 + o(1))) , which linearizes system (13). Furthermore, when Y = 0, the expression for X must be divisible by x. The first integrals of the linear system are then ψ1 = X −2 Ỹ 3 and ψ2 = Ỹ Z −1 . Pulling back these two first integrals we see that the initial system admits two independent analytic fist integrals of the desired form φ1 = ψ1 = x y 3 (1 + . . .) and φ2 = ψ1 ψ 2 −1 = y 2 z(1 + . . .). Since Ż = 2Z, Theorem 4 shows that the original system is in fact linearizable. Case 11 The system is ẋ = x(3 + ax − 2ey + cz), ẏ = y(−1 + ey + f z), ż = z(2 + gx + kz). 1 1 After the change of coordinates (X, Y, Z) = (x 2 , x 2 y, z) the system becomes   3 a 2 c Ẋ = X + X + z − eY, 2 2 2  1 a 2  c (14) Ẏ = Y + X + f+ z , 2 2 2 2 ż = z(2 + gX + kz). The singular point at the origin of system (14) is in the Poincaré domain. Since z = 0 is invariant, the two resonant monomials Y 4 and XY of the third equation must vanish. Additionally, for z = 0, a small calculation of the normal form of the two dimensional system shows that the resonant monomial Y 3 of the first equation must vanish. So system (14) is linearizable via an analytic change of coordinates which can be chosen as (X̃, Ỹ , Z̃) = (X − eY + o(X, Y, z), Y (1 + o(1)), z(1 + o(1))) . Furthermore, from the X ↔ −X symmetry of (14), the o(1) terms in Ỹ and Z̃ are even in X and the O(X, Y, z) term is odd in X. Also, when Y = 0, the expression for X̃ must be divisible by X. Thus, we can pull back the two first integrals ψ1 = X̃ −1 Ỹ 3 and ψ2 = X̃ −2 Ỹ 2 Z̃ of the linear system to obtain the two first integrals of the initial system φ1 = x y 3 (1 + o(1)) and φ2 = y 2 z(1 + o(1)). The expression for Z̃ in the original coordinates satisfies Ż = 2Z, and hence the system is also linearizable by Theorem 4. Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 17 Case 12 The system is ẋ = x(3 − 3dx − 3ey + cz), ẏ = y(−1 + dx + ey + f z), ż = z(2 − 8dx + kz). (15) We let ` = 1 − dx − ex and perform the change of variables (X, Y, Z) = (x, yz`−2 , z`−2 ). After a scaling by a factor of `, this brings the system to the form Ẋ = X(3 − ceY + cZ − cdXZ), Ẏ = Y (1 + (k + f )Z + e(f − k)Y + d(2c − f − k)XZ), (16) Ż = Z(2 − e(2f − k)Y + kZ + d(2c − k)XZ). The origin of this system is in the Poincaré domain. Since X = 0 is invariant for system (16) the two resonant monomials Y 3 and Y Z must vanish in the first equation. Additionally, Z = 0 is also invariant for system (16) and so the resonant monomial Y 2 in the last equation must vanish. Hence, for system (16) there is a linearising transformation of the form (X̃, Ỹ , Z̃) = (X(1 + o(1)), Y (1 + o(1)), Z(1 + o(1))). The first integrals φ1 = X̃ Ỹ 3 Z̃ −3 and φ2 = Ỹ 2 Z̃ −1 pull back to first integrals of the original system of the form φ1 = xy 3 (1 + . . .) and φ2 = y 2 z(1 + . . . ). Case 13 The system is ẋ = x(3 + ax + cz), ẏ = y(−1 + dx + ey), ż = z(2 + gx − ey + kz). The change of coordinates Y = yz transform the system above into  ẋ = x(3+ax+cz), Ẏ = Y 1+(d+g)x+kz , ż = z(2+gx+kz)−eY. (17) Note that the origin of system (17) is in the Poincaré domain. Since x = 0 is invariant for system (17), the resonant monomials Y Z and Y 3 in the first equation should vanish. Moreover, for x = 0 a small calculation of the normal form for the two dimensional system shows that the resonant monomial, Y 2 , should vanish in the last equation. Thus system (17) is linearizable by an analytic change of coordinates (X, Ỹ , Z) = (x(1 + o(1)), Y (1 + o(1)), z − eY + o(x, Y, z)). We can now construct two first integrals for the linear system φ = X −1 Ỹ 3 and ψ = X −1 Ỹ Z. Pulling back these two first integrals, we get two independent first integrals of the initial system and are of the desired form φ1 = φ2 ψ −3 = x y 3 (1 + o(1)) and φ2 = φψ −1 = y 2 z(1 + o(1)). Theorem 4 can then be applied to show the linearizability of the system from Ẋ = 3X. Case 14 The system is ẋ = x(3 + ax + by + cz), ẏ = y(−1 + dx − by), ż = z(2 + gx + by + kz). 18 W. Aziz, C. Christopher, J. Llibre and C. Pantazi The change of coordinates (X, Z) = (x/(1 + by)2 , z/(1 + by)) brings the system into the form Ẋ = X(3 + aX(1 + by) + cZ − 2bdXy)(1 + by), ẏ = y(−1 + dX(1 + by))(1 + by), (18) Ż = Z(2 + gX(1 + by) + kZ − bdXy)(1 + by). After rescaling the system by (1 + bY ) and applying the change of variable Y = Xy, system (18) becomes Ẋ = X(3 + aX + (ab − 2bd)Y + cZ), Ẏ = Y (2 + (a + d)X + (ab + bd)Y + cZ), (19) Ż = Z(2 + gX + (gb − bd)Y + kZ). Now the origin of system(19) is in the Poincaré domain and it is easy to see that there are no resonant terms. Hence, system (19) is linearizable via an analytic change of coordinates, (X̃, Ỹ , Z̃) = (X(1 + o(1)), Y (1 + o(1)), Z(1 + o(1))). The desired first integrals are φ1 = X̃ −2 Ỹ 3 = xy 3 (1 + . . .) and φ2 = X̃ −2 Y˜2 Z̃ = y 2 z(1 + . . .), and the linearizability follows from Theorem 4 since Ẋ = 3X, with X = x(1 + . . .). Case 15 The system is ẋ = x(3 + ax + 2hy + cz), ẏ = y(−1 − hy), ż = z(2 + gx + hy + kz). We use the change of variable (X, Z) = (x/(3(1 + by)), z/(2(1 + by))) and rescale the resulting system by 1 + hy we get     a k Ẋ = X 3 + X + 2cZ , ẏ = −y, Ż = Z 2 + 3gX + Z . 3 2 Clearly the first and third equation give a linearizable node and therefore the system is linearizable, and the original system must be integrable. Since Y = y/(1 + hy) satisfies Ẏ = −Y , the original system is linearizable by Theorem 4. Case 16 The system is ẋ = x(3 + ax + by + cz), ẏ = y(−1 + ax + by), ż = z(2 − ax − by + kz). Performing the change of coordinates (X, Y, Z) = (xz, yz, z) the system becomes Ẋ = X(5 + (c + k)Z), Ẏ = Y (1 + kZ), Ż = Z(2 + kZ) − aX − bY. (20) The origin of system (20) is in the Poincaré domain. Since X = 0 is invari- ant for (20), the resonant monomials Y Z 2 , Y 3 Z and Y 5 in the first equa- tion should vanish. For X = 0, a small calculation of the normal form for the two dimensional system shows that the resonant monomial, Y 2 , in the last equation vanishes. Thus, system (20) is linearizable via a transformation Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 19 (X̃, Ỹ , Z̃) = (X(1 + o(1)), Y (1 + o(1)), Z + a3 X − bY + o(X, Y, Z)). The first integrals of the linearized system, φ1 = X̃ Ỹ 3 Z̃ −4 and φ2 = Ỹ 2 Z̃ −1 , then pull back to integrals of the desired form φ1 = x y 3 (1 + . . .) and φ2 = y 2 z(1 + . . .). Theorem 4 can then be used to prove linearizability from the fact that Z̃ = z(1 + . . .) satisfies Z̃˙ = Z̃. Case 17 ẋ = x(3 + ax − 9f z), ẏ = y(−1 + dx + ey + f z), ż = z(2 + gx − 2ey − 2f z). 1 3 1 3 The change of variables (X, Y ) = (x 2 `− 2 , y x 2 `− 2 ), where ` = 1 − ey − f z, brings the system, after a scaling by ` to the form   3 3˜ 1 ˜2 Ẋ = X + `(deY + f gzX) + a` , 2 2 2   1 3˜ 1 (21) Ẏ = Y + `(deY + f gzX) + ( a + d)`˜2 , 2 2 2   ˜2 ż = z 2 + g ` , where `˜ = X − eY − f Xz. The origin of the transformed system (21) is in the Poincaré domain. Since X = 0 is invariant for the system (21), we conclude that the resonant monomial Y 3 in the first equation should vanish. Moreover, z = 0 is invariant for system (21), and so the resonant monomials Y 4 and XY must also vanish in the last equation. So for system (21) there is a linearizing change of coordinates (X̃, Ỹ , Z̃) = (X(1 + o(1), Y (1 + o(1)), z(1 + o(1))). Furthermore, from the form of the system (21), the terms of the series X̃ (resp. Ỹ and Z̃) are odd (resp. even) in the combined degree of X and Y . Thus the first integrals of (21), φ1 = X̃ −1 Ỹ 3 and φ2 = X̃ −2 Ỹ 2 Z̃, pull back to first integrals of the original system of the required form: φ1 = xy 3 (1 + o(1)) and φ2 = y 2 z(1 + o(1)). Case 18 We obtain the system ẋ = x(3+ax−ey−7f z), ẏ = y(−1+dx+ey+f z), ż = z(2+gx−2ey−2f z). A change of coordinate X = x `−2 where ` = 1 − ey − f z transforms the above system into Ẋ = X(3 + aX` + 2X(edy + f gz)) `, ẏ = y(−1 + dX`) `, ż = Z(2 + gX`)`. (22) We rescale the system by ` and perform the change of variables Y = Xy to give Ẋ = X(3 + aX − (2d − a)eY + (2g − a)f Xz), Ẏ = Y (2 + (a + d)X + (d − a)eY + (2g − d − a)f Xz), (23) ż = z(2 + gX − geY − gf Xz). The origin of (23) is in the Poincaré domain and it is easy to see that there are no resonant monomials. Hence system (23) is linearizable by a transformation 20 W. Aziz, C. Christopher, J. Llibre and C. Pantazi (X̃, Ỹ , Z̃) = (X(1 + o(1)), Y (1 + o(1)), z(1 + o(1))). The first integrals in the desired forms are φ1 = X̃ −2 Ỹ 3 = xy 3 (1 + . . .) and φ2 = X̃ −2 Y˜2 Z̃ = y 2 z(1 + . . .). Case 19 In this case system (1) can be written as ẋ = x(3 + ax + by + cz), ẏ = y(−1 + dx − by − cz), ż = z(2 + gx − by − cz). Under the change of coordinates (X, Y, Z) = (x, xy, xz) we obtain the system Ẋ = X(3+aX)+bY +cZ, Ẏ = Y (2+(a+d)X), Ż = Z(5+(a+g)X), (24) and the singular point is in the Poincaré domain. Since Z = 0 is invariant for system (24), the resonant monomial XY in the last equation must vanish. Thus, a change of coordinates (X̃, Ỹ , Z̃) = (X + bY − 2c Z + o(X, Y, Z), Y (1 + o(1)), Z(1 + o(1))) linearizes system (24). The first integrals of the linear system are then φ1 = X̃ −2 Ỹ 3 and ψ2 = −3 2 X̃ Ỹ Z̃, which pull back to first integrals in the desired form φ1 = φ = 3 x y 3 (1 + . . .) and φ2 = φ 2 ψ = y 2 z(1 + . . .). Since Z̃ = z(1 + . . .) satisfies Z̃˙ = Z̃, the system is linearizable from Theorem 4. Case 20 The system in this case becomes ẋ = x(3 + ax − 2hy − 2kz), ẏ = y(−1 + hy + kz), ż = z(2 + gx + hy + kz). 1 1 1 After the change of coordinates (X, Y, Z) = (x 2 , x 2 y, x 2 z), we get the system       3 a 2 1 a 2 7 a  Ẋ = X + X −hY −kZ, Ẏ = Y + X , Ż = Z + + g X2 . 2 2 2 2 2 2 (25) The origin of the resulting system is in the Poincaré domain. Since Z = 0 is invariant for system (25), we see that the resonant monomials Y 7 , XY 4 and X 2 Y , must vanish in the last equation. Also, for Z = 0, a small calculation of the normal form for the two dimensional system shows that the resonant monomial Y 3 in the first equation must also vanish. Thus, there is a trans- formation (X̃, Ỹ , Z̃) = (X − hY + k2 Z + o(X, Y, Z), Y (1 + o(1)), Z(1 + o(1))) which linearizes system (25). Furthemore, since the system is symmetric under X 7→ −X, the ex- pressions for X̃ (resp. Ỹ and Z̃) are odd (resp. even) in X. Thus the first integrals of the linear system φ1 = X̃ −1 Ỹ 3 and φ2 = X̃ −2 Ỹ 2 Z̃ pull back to first integrals φ1 = x y 3 (1 + . . .) and φ2 = y 2 z(1 + . . .) showing the system is integrable. Taking ξ = X̃ 2 = x(1 + . . .) we can apply Theorem 4 to show that the original system is linearizable.  Remark 3. According to Theorem 4 of [6], when the cofactors of x, y and z and the divergence of (1) are linearly independent, then the origin is lineariz- able if and only if it is integrable. However, since this condition only holds Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 21 on an open set in general, we have preferred to use alternative methods here to establish the linearizability. Acknowledgments J. Llibre is partially supported by the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. J. Llibre and C. Pantazi are also supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER). C. Pantazi is additionally partially supported by the Catalan Grant 2017SGR1049 and the Spanish MINECO-FEDER Grant PGC2018-098676-B-I00/AEI/FEDER/UE. References [1] J. Alavez–Ramı́rez, G. Blé, V. Castellanos and J. Llibre, On the global flow of a 3–dimensional Lotka–Volterra system, Nonlinear Anal. 75 (2012) 4114- 4125. [2] M.A. Almeida, M.E. Magalhães and I.C. Moreira, Lie symmetries and invariants of the Lotka–Volterra system, J. Math. Phys 36 (1995) 1854–1867. [3] M. Audin, Spinning tops. A course on integrable systems, 1996, Cambridge University Press, Cambridge. [4] W. Aziz, Analytic and Algebraic Aspects of Integrability for First Order Partial Differential Equations, Ph.D. Thesis, Plymouth University, 2013. [5] W. Aziz, Integrability and Linearizability of Three Dimensional Vector Fields, Qual. Theory Dyn. Syst. 13 (2014) 197–213. [6] W. Aziz and C. Christopher, Local Integrability and Linearizability of Three-dimensional Lotka-Volterra systems, Applied Mathematics and Com- putation, 219, (2012) 4067–4081. [7] L. R. Berrone and H. Giacomini, Inverse Jacobi Multiplier, Rendiconti Del Circolo Matematico Di Palermo, 3 (2003) 77–103. [8] M. Bobienski and H. Zoladek,The three–dimensional generalized Lotka– Volterra systems, Ergod. Th. & Dynam. Sys. 25 (2005) 759–791. [9] T.C. Bountis, A. Ramani, B. Grammaticos and B. Dorizzi, On the com- plete and partial integrability of non–Hamiltonian systems, Phys. A 128 (1984), 268–288. [10] A. D. Bruno, Analytical form of differential equations, Transactions of the Moscow Mathematical Society 25, (1971) 131–288. [11] A. D. Bruno, Local Methods in Nonlinear Differential Equations, Springer Verlag, New York, 1989. [12] L. Brenig, Complete factorisation and analytic solutions of generalized Lotka– Volterra equations, Phys. Lett. 133 (1988) 378–382. [13] L. Brenig and A. Goriely, Universal Canonical forms for time–continous dynamical systems, Phys. Rev. A 40 (1989) 4119–4122. 22 W. Aziz, C. Christopher, J. Llibre and C. Pantazi [14] C. A. Buzzi, R. A. T. Santos and J. Llibre, Final evolutions of a class of May-Leonard Lotka-Volterra systems, J. of Nonlinear Mathematical Physics, 27 (2020) 267–278. [15] L. Cairó, Darboux First Integral Conditions and Integrability of the 3D Lotka- Volterra System, J. of Nonlinear Mathematical Physics, 9 (2000) 511–531. [16] L. Cairó and J. Llibre, Darboux integrability for 3D Lotka-Volterra systems, J. of Phys. A: Math and Gen. 33 (2000) 2395–2406. [17] F. Cantrijn and W. Sarlet, Generalizations of Noether’s theorem in clas- sical mechanics, SIAM Rev. 23 (1981) 467–494. [18] Y. T. Christodoulides and P. A. Damianou, Darboux Polynomials for Lotka-Volterra Systems in Three Dimensions, J. of Nonlinear Mathematical Physics, 16 (2009) 339–354. [19] C. Christopher and C. Rousseau, Normalizable, Integrable and Linearizable Saddle Points in the Lotka-Volterra System, Qualitative Theory of Dynamical Systems, 5 (2004) 11–61. [20] C. Christopher, P. Mardes̆ić and C. Rousseau, Normalizability, Syn- chronicity, and Relative Exactness for Vector Fields in C2 , Qualitative Theory of Dynamical Systems, 10 (2004) 501–525. [21] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math. 2ème série 2 (1878), 60–96; 123–144; 151–200. [22] G. Darboux, De l’emploi des solutions particulières algébriques dans l’intégration des systèmes d’équations différentielles algébriques, C. R. Math. Acad. Sci. Paris 86 (1878), 1012–1014. [23] M. Dukarić, R. Oliveira and V. G. Romanovski, Local Integrability and Linearizability of a (1 : −1 : −1) Resonant Quadratic System, J. Dyn. Diff. Equat. 27 (2015) 1–17. [24] F. Dumortier, J. Llibre and J.C. Artés, Qualitative theory of planar dif- ferential systems, Springer–Verlag, 2006. [25] S. Gravel and P. Thibault, Integrability and linearizability of the Lotka- Volterra system with a saddle point with rational hyperbolicity ratio, J. Differ- ential Equations, 184 (2002) 20–47. [26] A. Goriely, Integrability and non integrability of dynamical systems, Advances Series in Nonlinear Dynamics 19 World Scientific PublishingCo., Inc., River Edge, NJ,2001. [27] W. E. Lamb, Theory of an optical maser, Phys. Rev. A 134 (1964) 1429–1450. [28] G. Laval and R. Pellat, Plasma Physics, Proc. Summer School of Theoret- ical Physics (1975) (New York: Gordon and Breach). [29] P.D. Lax, Integrals of Nonlinear Equations of Evolution and Solitary Waves, Commun. Pure Appl. Math. 21 (1968), 467–490. [30] A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Natl. Acad. Sci. USA 6 (1920) 410–415. [31] J. Llibre and C. Valls, Polynomial, rational and analytic first integrals for a family of 3–dimensional Lotka–Volterra systems, Z. Angew. Math. Phys. 62 (2011) 761–777. Three dimensional Lotka-Volterra Systems with 3:-1:2-resonance 23 [32] J. Llibre and X. Zhang, Darboux theory of integrability in Cn taking into account the multiplicity, J. Differential Equations, 246 (2009) 541–551. [33] J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in Rn taking into account the multiplicity at infinity, Bull. Sci. Math. 133 (2009) 765–778. [34] J. Llibre and X. Zhang, Dynamics of Some Three-Dimensional Lotka- Volterra Systems, Mediterr. J. Math. 14(126) (2017), 13 pp. [35] Y. Ilyashenko and S. Yakovenko, Lectures on Anylitic Differential Equa- tions, American Mathematical Society, 2007. [36] R.S. Maier, The integration of three-dimensional LotkaVolterra systems, P. Roy. Soc. A-Math. Phy. 469:20120693, (2013) 27 pp. [37] R. M. May, Stability and Complexity in Model Ecosystems, (Princeton, NJ: Princeton University Press), 1974. [38] R. M. May and W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl.Math. 29 (1975) 243–256. [39] A.C. Murza and A.E. Teruel, Global dynamics of a family of 3–D Lotka– Volterra systems, Dynamical Systems: An International Journal, 25 (2010) 269– 284. [40] V. W. Noonburg, A neural network modeled by an adaptive Lotka–Volterra system, SIAM J. Appl. Math. 49 (1989) 1779. [41] P.J. Olver, Applications of Lie groups to differential equations, Springer, New York, 1986. [42] G.M. Pfister, G. Schönemann, H. Decker and W. Greuel, Singu- lar 2-0-3 – A computer algebra system for polynomial computations, 2002. http://www.singular.uni-kl.de. [43] V. A. Pliss, On the reduction of an analytic system of differential equations to linear form Differential Equations, 1 (1965) 153–161. [44] M.F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992) 673–688. [45] P. Van den Driessche and M.L. Zeeman, Three–dimensional competitive Lotka–Volterra systems with no periodic orbits, Siam J.Appl.Math. 58 (1998) 227–234. [46] V. Volterra, Leçons sur la Theéorie Mathématique de la Lutte pour la vie, Paris: Gauthier Villars, 1931. [47] S. Walcher, Symmetries and Convergence of normal form transformations. Monografias de la Real Academia de Ciencias de Zaragoza, 25 (2004) 251–268. [48] J.-A. Weil, First integrals and Darboux polynomials of homogeneous linear differential systems, Applied algebra, algebraic algorithms and error-correcting codes (Paris, 1995), 469–484, Lecture Notes in Comput. Sci., 948 Springer, Berlin, 1995. [49] M. L. Zeeman, Hopf bifurcations in competitive three dimensional Lotka– Volterra systems, Dynam. Stability Systems, 8 (1993) 189–217. 24 W. Aziz, C. Christopher, J. Llibre and C. Pantazi Waleed Aziz1 , Colin Christopher2 , Jaume Llibre3 and Chara Pantazi4 1 Department of Mathematics, College of Science, Salahaddin University-Erbil, Kurdistan Region-Iraq Information Technology Department, Faculty of Science, Tishk International Uni- versity, Erbil, KRG, Iraq e-mail: waleed.aziz@plymouth.ac.uk 2 School of Engineering, Computing and Mathematics, Plymouth University, Ply- mouth UK, PL4 8AA e-mail: C.Christopher@plymouth.ac.uk 3 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Catalonia, Spain e-mail: jllibre@mat.uab.cat 4 Departament de Matemàtiques, Universitat Politècnica de Catalunya, (EPSEB), Av. Doctor Marañón, 44–50, 08028 Barcelona, Spain e-mail: chara.pantazi@upc.edu