  Let $(P_n)_n$ and $(Q_n)_n$ be two sequences of monic polynomials linked by a
type structure relation such as $$
Q_{n}(x)+r_nQ_{n-1}(x)=P_{n}(x)+s_nP_{n-1}(x)+t_nP_{n-2}(x)\;, $$ where
$(r_n)_n$, $(s_n)_n$ and $(t_n)_n$ are sequences of complex numbers.
  First, we state necessary and sufficient conditions on the parameters such
that the above relation becomes non-degenerate when both sequences $(P_n)_n$
and $(Q_n)_n$ are orthogonal with respect to regular moment linear functionals
${\bf u}$ and ${\bf v}$, respectively.
  Second, assuming that the above relation is non-degenerate and $(P_n)_n$ is
an orthogonal sequence, we obtain a characterization for the orthogonality of
the sequence $(Q_n)_n$ in terms of the coefficients of the polynomials $\Phi$
and $\Psi$ which appear in the rational transformation (in the distributional
sense) $\Phi {\bf u}=\Psi {\bf v}\; .$
  Some illustrative examples of the developed theory are presented.
