A classification of isolated singularities of elliptic Monge-Amp\`ere equations in dimension two

Let ${\mathcal{M}}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Amp`ere equation $\mathrm{d}\mathrm{e}\mathrm{t}(D^{2}z)=\phi (x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that ${\mathcal{M}}_1$ is in one-to-one correspondence with $$\mathcal{M}$_2\times Z_2$,where$$\mathcal{M}$_2$ is a suitable subset of the class of regular, real analytic strictly convex Jordan curves in $R^2$. We also describe the asymptotic behavior of solutions of the Monge-Amp`ere equation in the $C^k$-smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation ${\rm det} (D^2 z +$\mathcal{A}$(x,y,z,Dz))=\varphi(x,y,z,Dz)>0$.