  We extend the Eruguin result exposed in the paper "Construction of the whole
set of ordinary differential equations with a given integral curve" published
in 1952 and construct a differential system in $\Bbb{R}^N$ which admits a given
set of the partial integrals, in particular we study the case when theses
functions are polynomials. We construct a non-Darboux integrable planar
polynomial system of degree $n$ with one invariant irreducible algebraic curve
$g(x,y)=0$. For this system we analyze the Darboux integrability, Poincare's
problem and 16th's Hilbert problem for algebraic limit cycles.
