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Consider a planar polynomial vector field $X$, and assume it admits asymbolic first integral $\mathcal{F}$, i.e. of the $4$ classes, in growingcomplexity: Rational, Darbouxian, Liouvillian and Riccati. If $\mathcal{F}$ isnot rational, it is sometimes possible to reduce it to a simpler class firstintegral. We will present algorithms to reduce symbolic first integral to alower complexity class. These algorithms allow to find the minimal class firstintegral and in particular to test the existence of a rational first integralexcept in the case where $\mathcal{F}$ is a $k$-Darbouxian first integralwithout singularities and $k\in\{2,3,4,6\}$. In this case, several examples arebuilt and a procedure is presented which however requires the computation ofelliptic factors in the Jacobian of a superelliptic curve.

Reduction of symbolic first integrals of planar vector fields