Invariance of the generalized oscillator under a linear transformation of the related system of orthogonal polynomials

We consider the families of polynomials $\mathbb P=\{P_n(x)\}_{n=0}^\infty$ and $\mathbb Q=\{Q_n(x)\}_{n=0}^\infty$ orthogonal on the real line with respect to the respective probability measures $\mu$ and $\nu$. We assume that $\{Q_n(x)\}_{n=0}^\infty$ and $\{P_n(x)\}_{n=0}^\infty$ are connected by linear relations. In the case $k=2$, we describe all pairs $(\mathbb P,\mathbb Q)$ for which the algebras $\mathfrak A_P$ and $\mathfrak A_Q$ of generalized oscillators generated by $\{Q_n(x)\}_{n=0}^\infty$ and $\{P_n(x)\}_{n=0}^\infty$ coincide. We construct generalized oscillators corresponding to pairs $(\mathbb P,\mathbb Q)$ for arbitrary $k\ge1$.