Orthogonal polynomial solutions in two variables satisfying the second order partial differential equation with polynomial coefficients \[ L[u]\equiv Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}=\lambda u\tag{*} \] are considered [see \textit{H. L. Krall} and \textit{I. M. Sheffer}, Ann. Mat. Pura Appl., IV. Ser. 76, 325-376 (1967; Zbl 0186.38602)]. It is proved that if the differential equation (*) has an orthogonal polynomial system as solutions, then \(B^2\not=AC\) in any nonempty open subset of \(\mathbb R^2\) (nonparabolic case of (*)). In addition it is shown that if the equation (*) has an orthogonal polynomial system as solutions, then the differential operator \(L\) must be symmetrizable [see \textit{L. L. Littlejohn}, Proc. Int. Symp., Segovia/Spain 1986, Lect. Notes Math. 1329, 98-124 (1988; Zbl 0653.42022)]. Introducing a weak orthogonal polynomial system of solutions to (*) the Rodrigues type formula was obtained too. To illustrate the results obtained some examples are given.