On the polynomial differential systems having polynomial first integrals (Q414797)

Consider the two-dimensional system NEWLINE\[NEWLINE \dot{x}=P(x,y),\quad \dot{y}=Q(x,y),NEWLINE\]NEWLINE where \(P\) and \(Q\) are complex polynomial functions in \(x\) and \(y\). The authors study the existence of polynomial first integrals for the subclass of systems such that their minimal polynomial first integrals have a unique critical remarkable value. Some general properties of such first integrals are given. The authors give a complete characterization of all Liénard polynomial systems NEWLINE\[NEWLINE\dot{x}=y,\;\dot{y}=-f(x)y-g(x)NEWLINE\]NEWLINE having a minimal polynomial first integral with a unique critical remarkable value.