Bivariate version of the Hahn-Sonine theorem

We consider orthogonal polynomials in two variables whose derivatives with respect to $x$ are orthogonal. We show that they satisfy a system of partial differential equations of the form where $\deg\alpha\leq2$, $\deg\beta\leq1$, $\overrightarrow{U} _{n}=(U_{n0},U_{n-1,1},\cdots,U_{0n})$ is a vector of polynomials in $x$ and $y$ for $n\geq0$, and $\Lambda_{n}$ is an eigenvalue matrix of order $(n+1)\times(n+1)$ for $n\geq0$. Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran's.