  We study algebraic integrability of complex planar polynomial vector fields
$X=A (x,y)(\partial/\partial x) + B(x,y) (\partial/\partial y) $ through
extensions to Hirzebruch surfaces. Using these extensions, each vector field
$X$ determines two infinite families of planar vector fields that depend on a
natural parameter which, when $X$ has a rational first integral, satisfy strong
properties about the dicriticity of the points at the line $x=0$ and of the
origin. As a consequence, we obtain new necessary conditions for algebraic
integrability of planar vector fields and, if $X$ has a rational first
integral, we provide a region in $\mathbb{R}_{\geq 0}^2$ that contains all the
pairs $(i,j)$ corresponding to monomials $x^i y^j$ involved in the generic
invariant curve of $X$.

Algebraic Geometry;Dynamical Systems