Spaces of matrices with few eigenvalues (II)

Let $F$ be a field, and $\mathcal{M}$ be a linear subspace of $n$-by-$n$ matrices with entries in $F$ that have at most two eigenvalues in $F$ (respectively, at most one non-zero eigenvalue in $F$). In a previous article, we have determined the greatest possible dimension for $\mathcal{M}$ when the characteristic of $F$ is not $2$. In this article and its sequel, we solve this problem for all fields with characteristic $2$.