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Let $(P_n)_n$ and $(Q_n)_n$ be two sequences of monic polynomials linked by atype 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 suchthat 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 $(P_n)_n$ isan orthogonal sequence, we obtain a characterization for the orthogonality ofthe sequence $(Q_n)_n$ in terms of the coefficients of the polynomials $\Phi$and $\Psi$ which appear in the rational transformation (in the distributionalsense) $\Phi {\bf u}=\Psi {\bf v}\; .$  Some illustrative examples of the developed theory are presented.

Orthogonal polynomials generated by a linear structure relation: Inverse problem