\aocpm,\,
\Pnkm{k}{\pm}{l}, \Pnkm{k}{l}{\pm},\, \Pnkm{k,\,l}{m}{n}.$$ It is assumed that indexes $k,\,l,\,m,\,n\geq0$ are fulfilled one of the following conditions $$\begin{aligned}
1)\qquad & m+n=k+l,\quad m-n=k-l\pm 2t;\nn\\
2)\qquad & m-n=k-l,\quad m+n=k+l\pm 2t; \label{13-6.2}\\
3)\qquad & kl=0,\quad mn=0;\nn\end{aligned}$$ and the generators satisfy the commutation relations - , - , - , - .

We determine the algebra $\AG$ of generalized Chebyshev - Koornwinder oscillator as infinite-dimensional Lie algebra which is a minimal extension of the algebra $\widetilde{\AG}$. To define the algebra $\AG$ we consider first the algebra $\widehat{\AG}$ obtained from $\widetilde{\AG}$ by addition of all formal series in generators . The commutation relations of the algebra $\widehat{\AG}$ induced by the relations - , - , - , - . It is helpful to introduce the following notation for some of such series, $$\label{13-6.3}
\widehat{P}^{[k],\,l}_{[m],\,n}=\bigoplus_s  P^{s+k,\,l}_{s+m,\,n},\qquad
\widehat{P}^{k,\,[l]}_{m,\,[n]}=\bigoplus_s P^{k,\,s+l}_{m,\,s+n}.$$

[**Remarks**]{} 1) Recall that $\Pnkm{m,\,n}{k}{l}=0$, if at least one of the indices is negative;

2)Operators , , can be expressed by the operators as follows: $$\begin{aligned}
\Pnck{n}{k}&=\hPnkm{[n],\,k}{[k]}{n};
\quad\Pnkc{n}{k}=\hPnkm{k,\,[n]}{n}{[k]};
\\[9pt]
\Pnkm{k}{+}{l}& =\hPnkm{[k],\,k}{[l]}{l};
\quad\Pnkm{k}{-}{l}=\Pnck{(k)}{l}=\hPnkm{[k],\,l}{[l]}{k};
\\[9pt]
\Pnkm{k}{l}{+}& =\hPnkm{k,\,[k]}{l}{[l]};
\quad\Pnkm{k}{l}{-}=\hPnkm{l,\,[k]}{k}{[l]};
\quad\acopm  =\hPnkm{[0],\,0}{[\pm 1]}{0};\quad
\aocpm =\hPnkm{0,\,[0]}{0}{[\pm 1]}.\end{aligned}$$

The following commutation relations $$\label{13-6.12}
\left\{
\begin{aligned}
& [\mathbb{N}_1,\mathbb{N}_2]=0,\quad [\mathbb{N}_1,\aspm]=\mp\aspm,\quad [\mathbb{N}_1,\arm]=\pm\arpm;\\[3pt]
& [\mathbb{N}_1,\widehat{P}^{[k],\,l}_{[m],\,n}]=(m-k)\widehat{P}^{[k],\,l}_{[m],\,n};\\[5pt]
& [\mathbb{N}_1,\widehat{P}^{k,\,[l]}_{m,\,[n]}]=(m-k)\widehat{P}^{k,\,[l]}_{m,\,[n]};\\[3pt]
& [\mathbb{N}_1,\Pnkm{k,\,l}{n}{m}]=(n-k)\Pnkm{k,\,l}{n}{m};
\end{aligned}
\right.$$ $$\label{13-6.13}
\left\{
\begin{aligned}
&  [\mathbb{N}_2,\aspm]=\pm\aspm,\quad [\mathbb{N}_2,\arpm]=\pm\arpm,\quad
[\mathbb{N}_2,\Pnkm{k,\,l}{n}{m}]=(m-l)\Pnkm{k,\,l}{n}{m};\\[3pt]
