Abstract

In this talk we will study some aspects of the integrability problem for polynomial vector fields $x\prime =P(x,y)$, $y\prime =Q(x,y)$. Firstly, we will remember the Darboux integrability theory and their recents improvements. We will introduce the Generalized integrability theory for non-algebraic invariant curves from the definition of generalized cofactor. Finally, we will analize the possible existence of first integrals of the form $I(x,y)=h(x)(y-g$₁(x))^a₁(y-g₂(x))^a₂...(y-g_k(x))^a_k, where $g$_i(x) are particular solutions of $dy/dx=Q(x,y)/P(x,y)$, $a$_i are arbitrary constants and $h(x)$ is an arbitrary function. We will show that for certain systems some of the particular solutions remain arbitrary and the other ones are explicitly determined or are fuctionally related to the arbitrary solutions. We will obtain in this way a nonlinear superposition principle that generalize the classical nonlinear superposition principle of the Lie theory. In these cases we will obtain an expression of $I(x,y)$ that we call a generalized nonlinear superposition principle.