  In present paper we propose an approach based on examination of the structure
of the general solution of equations of the type dy/dx=P(x,y)/Q(x,y), with P
and Q polynomials only in y. Under the term structure we mean the dependency
character of solution from arbitrary constant. We describe a common form of the
structures for foregoing equations. In such a way one can obtain a
differential-algebraic polynomial system for undetermined parameters of the
structures. The successful solution of this system automatically leads to
finding the general solution of ODE's. We demonstrate on examples that proposed
method gives, as a first step, the systematic way for obtaining new
integrability conditions and general solutions for various families of rational
first-order ODE's.
