Quasi-monomials with respect to subgroups of the plane affine group

Let $H$ be a subgroup of the plane affine group ${\rm Aff}(2)$ considered with the natural action on the vector space of two-variable polynomials. The polynomial family $\{ B_{m,n}(x,y) \}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $ \{ x^m y^n \} $ and $\{ B_{m,n}(x,y) \}$ have \textit{identical} matrices. We obtain a criterion of quasi-monomiality for the case when the group $H$ is generated by rotations and translations in terms of exponential generating function for the polynomial family $\{ B_{m,n}(x,y) \}$.