  Let $\mathbb{F}$ be a field of characteristic different from $2$ and $3$, and
let $V$ be a vector space of dimension $2$ over $\mathbb{F}$. The generic
classification of homogeneous quadratic maps $f\colon V\to V$ under the action
of the linear group of $V$, is given and efficient computational criteria to
recognize equivalence are provided.
