If $mathcal{M}=(M, abla)$ 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 $tilde{mathcal{M}}=(M,tilde abla)$ 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 $ abla=tilde abla$ 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.
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