\left(a^{+}_{0,\,\bullet}a^{-}_{0,\,\bullet}+a^{-}_{0,\,\bullet}a^{+}_{0,\,\bullet}\right).\end{aligned}$$ The Hamiltonian $H_0$ is a bounded selfadjoint operator in $\H$. ChK-polynomials $U_{k,\,l}(z,\,\z)$ are eigenfunctions for $H_0$ with eigenvalues $$\mu_{0,\,0}=\frac{1}{5};\quad\mu_{0,\,N}=\mu_{N,\,0}=\frac{2}{5}, \,(\text{for}\, N\geq1),
\quad \mu_{k,\,l}=0, \,(\text{for}\, k,l\geq1).$$ Then the algebra $\mathfrak{A}_{0}$ of generalized boundary oscillator was defined in [@N46] as a closure of the associative algebra with generators $$\label{13-2.36}
\mathbb{I},\quad \mathbb{N}_1,\quad \mathbb{N}_2,\quad \Pnkm{s,\,0}{m}{0},
\quad \Pnkm{0,\,s}{0}{m},\quad \Pnkm{k,\,0}{0}{m},\quad \Pnkm{0,\,k}{m}{0},$$ satisfying the commutation relations $$\label{13-2.37}
\left\{
\begin{aligned}
\left[\mathbb{N}_1,\mathbb{N}_2\right]=0,
\quad [\mathbb{N}_1,\Pnkm{k,\,0}{m}{0}]=(m-k)\Pnkm{k,\,0}{m}{0},\quad
[\mathbb{N}_1,\Pnkm{0,\,k}{0}{m}]=0,& \\
[\mathbb{N}_1,\Pnkm{k,\,0}{0}{n}]=-k\Pnkm{k,\,0}{0}{n},\quad
[\mathbb{N}_1,\Pnkm{0,\,l}{m}{0}]=m\Pnkm{0,\,l}{m}{0};\qquad\qquad&
\end{aligned}
\right.$$

$$\label{13-2.38}
\left\{
\begin{aligned}
\left[\mathbb{N}_2,\Pnkm{k,\,0}{m}{0}\right]=0,
\quad \left[\mathbb{N}_2,\Pnkm{0,\,k}{0}{m}\right]=(m-k)\Pnkm{0,\,k}{0}{m},& \\
\left[\mathbb{N}_2,\Pnkm{k,\,0}{0}{n}\right]=n\Pnkm{k,\,0}{0}{n},\quad
\left[\mathbb{N}_2,\Pnkm{0,\,l}{m}{0}\right]=-l\Pnkm{0,\,l}{m}{0};&
\end{aligned}
\right.$$

$$\label{13-2.39}
\left\{
\begin{aligned}
\left[\Pnkm{k,\,0}{m}{0},\Pnkm{l,\,0}{n}{0}\right]&=
\delta_{k,\,n}\Pnkm{l,\,0}{m}{0}-\delta_{m,\,l}\Pnkm{k,\,0}{n}{0}, \\
\left[\Pnkm{k,\,0}{m}{0},\Pnkm{0,\,l}{0}{n}\right]&=
\delta_{k,\,0}\delta_{n,\,0}\Pnkm{0,\,l}{m}{0}-
\delta_{m,\,0}\delta_{l,\,0}\Pnkm{k,\,0}{0}{n};
\end{aligned}
\right.$$

$$\label{13-2.40}
\left\{
\begin{aligned}
\left[\Pnkm{k,\,0}{m}{0},\Pnkm{l,\,0}{0}{n}\right]&=
\delta_{k,\,0}\delta_{n,\,0}\Pnkm{l,\,0}{m}{0}-
\delta_{l,\,m}\Pnkm{k,\,0}{0}{n}, \\
\left[\Pnkm{k,\,0}{m}{0},\Pnkm{0,\,l}{n}{0}\right]&=
\delta_{k,\,n}\Pnkm{0,\,l}{m}{0}-
\delta_{m,\,0}\delta_{l,\,0}\Pnkm{k,\,0}{n}{0};
\end{aligned}
\right.$$

$$\label{13-2.41}
\left\{
\begin{aligned}
\left[\Pnkm{0,\,k}{0}{m},\Pnkm{0,\,l}{0}{n}\right]&=
\delta_{k,\,n}\Pnkm{0,\,l}{0}{m}-\delta_{m,\,l}\Pnkm{0,\,l}{0}{n}, \\
\left[\Pnkm{0,\,k}{0}{m},\Pnkm{l,\,0}{0}{n}\right]&=
\delta_{k,\,n}\Pnkm{l,\,0}{0}{m}-
\delta_{m,\,0}\delta_{l,\,0}\Pnkm{0,\,k}{0}{n}, \\
\left[\Pnkm{0,\,k}{0}{m},\Pnkm{0,\,l}{n}{0}\right]&=
\delta_{k,\,0}\delta_{n,\,0}\Pnkm{0,\,l}{0}{m}-
\delta_{l,\,m}\Pnkm{0,\,k}{n}{0};
\end{aligned}
\right.$$

$$\label{13-2.42}
\left\{
\begin{aligned}
\left[\Pnkm{k,\,0}{0}{n},\Pnkm{0,\,l}{m}{0}\right]&=
\delta_{k,\,m}\Pnkm{0,\,l}{0}{n}-
\delta_{l,\,n}\Pnkm{k,\,0}{m}{0}, \\
\left[\Pnkm{k,\,0}{0}{n},\Pnkm{l,\,0}{0}{m}\right]&=
\delta_{k,\,0}\delta_{m,\,0}\Pnkm{l,\,0}{0}{n}-
\delta_{l,\,0}\delta_{n,\,0}\Pnkm{k,\,0}{0}{m}, \\
\left[\Pnkm{0,\,k}{n}{0},\Pnkm{0,\,l}{m}{0}\right]&=
\delta_{k,\,0}\delta_{m,\,0}\Pnkm{0,\,l}{n}{0}-
\delta_{l,\,0}\delta_{k,\,0}\Pnkm{0,\,k}{m}{0}.
\end{aligned}
\right.$$

Algebra of generalized sectorial oscillator
===========================================

In this section, we define the infinite-dimensional Lie algebra $\mathfrak{A}_s$ as a closure of the sectorial oscillator algebra which is the (minimal) expansion of an associative algebra determined and . To do this we need some auxiliary operators entered below. Recall that $$\label{13-3.1}
P^{m,\,n}_{s,\,t}U_{k,\,l}(z,\,\z)=\delta_{k,\,m} \delta_{l,\,n}U_{s,\,t}(z,\,\z),$$ if $m,n,s,t,k,l\geq0$, and $$P^{m,\,n}_{s,\,t}U_{k,\,l}(z,\,\z)=0,$$ if at least one of the indices is 