The Kowalevskaya exponents and rational integrability of polynomial differential systems

This paper primarily grows from the paper of Llibre and Zhang [J. Llibre, X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity 15 (2002) 1269–1280] with the following essential generalizations: (i) we prove that the link established in the mentioned paper between the Kowalevskaya exponents and the degree of the polynomial first integrals holds not only for (1,…,1)-2 type systems but also for any (s1,…,sn)-d type systems. (ii) by using different methods, we obtain necessary and sufficient conditions for planar (s1,s2)-d systems to have rational first integrals, whereas in the mentioned paper, only (s1,s2)-2 type systems and only polynomial integrability are considered.As an application of the methods and the results, we present an illustrative and well studied example to show its non-existence of polynomial first integrals.