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We consider two families of polynomials $\mathbb{P}=\polP$ and$\mathbb{Q}=\polQ$\footnote{Here and below we consider only monic polynomials.}orthogonal on the real line with respect to probability measures $\mu$ and$\nu$ respectively. Let $\polQ$ and $\polP$ connected by the linear relations$$ Q_n(x)=P_n(x)+a_1P_{n-1}(x)+...+a_kP_{n-k}(x).$$ Let us denote$\mathfrak{A}_P$ and $\mathfrak{A}_Q$ generalized oscillator algebrasassociated with the sequences $\mathbb{P}$ and $\mathbb{Q}$. In the case $k=2$we describe all pairs ($\mathbb{P}$,$\mathbb{Q}$), for which the algebras$\mathfrak{A}_P$ and $\mathfrak{A}_Q$ are equal. In addition, we constructcorresponding algebras of generalized oscillators for arbitrary $k\geq1$.

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