The paper deals with second-order partial differential equations \[ L[u]=A(x,y) u_{xx}+2 B(x,y)u_{xy}+ C(x,y)u_{yy}+ D(x,y)u_x+ E(x,y)u_y= \lambda_nu, \tag{*} \] which have orthogonal polynomial eigenfunctions. It is shown that these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. Using this result more examples of orthogonal polynomials satisfying differential equations of the type (*) are obtained, which do not appear in the classification by Krall and Sheffer.