Algebraic integrability with bounded genus

We provide an algorithm which decides whether a polynomial foliation ${\mathcal{F}}^{{\mathbb{C}}^2}$ on the complex plane has a polynomial first integral of genus $g\mathrm{\ne }1$. Except in a specific case, an extension of the algorithm also decides if ${\mathcal{F}}^{{\mathbb{C}}^2}$ has a rational first integral of that genus.