  If $\mathcal{M}=(M,\nabla)$ is an affine surface, let
$\mathcal{Q}(\mathcal{M}):=\ker(\mathcal{H}+\frac1{m-1}\rho_s)$ be the space of
solutions to the quasi-Einstein equation for the crucial eigenvalue. Let
$\tilde{\mathcal{M}}=(M,\tilde\nabla)$ be another affine structure on $M$ which
is strongly projectively flat. We show that
$\mathcal{Q}(\mathcal{M})=\mathcal{Q}(\tilde{\mathcal{M}})$ if and only if
$\nabla=\tilde\nabla$ and that $\mathcal{Q}(\mathcal{M})$ is linearly
equivalent to $\mathcal{Q}(\tilde{\mathcal{M}})$ if and only if $\mathcal{M}$
is linearly equivalent to $\tilde{\mathcal{M}}$. We use these observations to
classify the flat Type~$\mathcal{A}$ connections up to linear equivalence, to
classify the Type~$\mathcal{A}$ connections where the Ricci tensor has rank 1
up to linear equivalence, and to study the moduli spaces of Type~$\mathcal{A}$
connections where the Ricci tensor is non-degenerate up to affine equivalence.
