Non-existence criteria for Laurent polynomial first integrals

Abstract

In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial first integrals for a general nonlinear systems of ordinary differential equations $\dot x = f(x)$, $x \in \mathbb{R}^n$ with $f(0) = 0$. We show that if the eigenvalues of the Jacobi matrix of the vector field $f(x)$ are $\mathbb{Z}$-independent, then the system has no nontrivial Laurent polynomial integrals.