Classification of the non-trivial $2$-$(k^2,k,\lambda )$ designs, with $\lambda \mid k$, admitting a flag-transitive automorphism group of affine type

The pairs $(\mathcal{D},G)$, where $\mathcal{D}$ is a non-trivial $2$-$(k^2,k,\lambda )$ design, with $\lambda \mid k$, and $G$ is a flag-transitive automorphism group of $\mathcal{D}$ of affine type such that $G\nleqq A\mathrm{\Gamma }{L}_1(k^2)$, are classified.