  In this paper we contribute towards the classification of partially symmetric
tensors in $\mathbb{F}_q^3\otimes S^2\mathbb{F}_q^3$, $q$ even, by classifying
planes which intersect the Veronese surface $\mathcal{V}(\mathbb{F}_q)$ in at
least one point, under the action of $K\leq \rm{PGL}(6,q)$, $K\cong
\rm{PGL}(3,q)$, stabilising the Veronese surface. We also determine a complete
set of geometric and combinatorial invariants for each of the orbits.
