Reduction of symbolic first integrals of planar vector fields.

Consider a planar polynomial vector field $X$, and assume it admits a symbolic first integral $\mathcal{F}$, i.e. of the $4$ classes, in growing complexity: Rational, Darbouxian, Liouvillian and Riccati. If $\mathcal{F}$ is not rational, it is sometimes possible to reduce it to a simpler class first integral. We will present algorithms to reduce symbolic first integral to a lower complexity class. These algorithms allow to find the minimal class first integral and in particular to test the existence of a rational first integral except in the case where $\mathcal{F}$ is a $k$-Darbouxian first integral without singularities and $k\in\{2,3,4,6\}$. In this case, several examples are built and a procedure is presented which however requires the computation of elliptic factors in the Jacobian of a superelliptic curve.