  Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic,
real analytic Monge-Amp\`ere equation ${\rm det} (D^2 z)=\varphi(x,y,z,Dz)>0$
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 ${\rm det} (D^2 z
+\mathcal{A}(x,y,z,Dz))=\varphi(x,y,z,Dz)>0$.
