On non-integrability of general systems of differential equations (Q1909922)

The author establishes non-integrability conditions for the non-Hamiltonian analytic semi-quasihomogeneous differential equation \(u'= f(u)\), \(u\in \mathbb{C}^n\) in a neighborhood of the equilibrium point \(u= 0= f(0)\). These conditions for non-existence of smooth (or polynomial) integrals consist in the verification of some resonance relations between the eigenvalues of some associated (Kowalevsky's) matrix. It should be noted that the method of investigation used in the paper is based on some previous works of H. Yoshida on non-existence of an analytical integral in Hamiltonian systems. As examples, a two-dimensional system of Volterra-Lotka type, a perturbed Oregonator model and the Euler-Arnold equations are studied.