The Kowalevskaya exponents and rational integrability of polynomial differential systems

Abstract

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 ( s 1 , … , s n ) - d type systems. (ii) by using different methods, we obtain necessary and sufficient conditions for planar ( s 1 , s 2 ) - d systems to have rational first integrals, whereas in the mentioned paper, only ( s 1 , s 2 ) -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.