On the solution of the Monge–Ampere equation $Z_{xx}Z_{yy}-Z_{xy}^{2}=f(x,y)$ with quadratic right side

For the Monge–Ampere equation $Z_{xx}Z_{yy}-Z_{xy}^{2}=b_{20}x^{2}+b_{11}xy+b_{02}y^{2}+b_{00}$ we consider the question on the existence of a solution $Z(x,y)$ in the class of polynomials such that $Z=Z(x,y)$ is a graph of a convex surface. If $Z$ is a polynomial of odd degree, then the solution does not exist. If $Z$ is a polynomial of $4$-th degree and $4b_{20}b_{02}-b_{11}^{2}>0$, then the solution also does not exist. If $4b_{20}b_{02}-b_{11}^{2}=0$, then we have solutions.