of all formal series in generators , including operators . The commutation relations of the algebra $\widehat{\AG}_0$ are induced by the relations - . The relations - , as well as $$\begin{gathered}
[\mathbb{N}_1,\acopm]=\pm \acopm,\quad [\mathbb{N}_2,\acopm]=0,\quad
[\acop ,\acom]=-\Pnkm{0,\,0}{0}{0},\nn\\
[\Pnkm{k,\,0}{m}{0},\acopm]=\Pnkm{k\mp 1,\,0}{m}{0}-\Pnkm{k,\,0}{m\pm 1}{0},\quad
[\Pnkm{0,\,k}{0}{m},\acopm]=\delta_{k,\,0}\Pnkm{\mp 1,\,0}{0}{m}-
\delta_{m,\,0}\Pnkm{0,\,k}{\pm 1}{0},
\label{13-5.1}\\
[\Pnkm{k,\,0}{0}{m},\acopm]=\Pnkm{k\mp 1,\,0}{0}{m}-\delta_{m,\,0}\Pnkm{k,\,0}{\pm 1}{0},\quad
[\Pnkm{0,\,l}{n}{0},\acopm]=\delta_{l,\,0}\Pnkm{\mp 1,\,0}{n}{0}-\Pnkm{0,\,l}{n\pm 1}{0},\nn\end{gathered}$$ $$\begin{gathered}
[\mathbb{N}_1,\aocpm]=0,\quad [\mathbb{N}_2,\aocpm]=\pm \aocpm,\quad
[\aocp ,\aocm]=\Pnkm{0,\,0}{0}{0},\nn\\
[\Pnkm{k,\,0}{m}{0},\aocpm]=\delta_{k,\,0}\Pnkm{0,\,\mp 1}{m}{0}
-\delta_{m,\,0}\Pnkm{k,\,0}{0}{\pm 1},\quad
[\Pnkm{0,\,k}{0}{m},\aocpm]=\Pnkm{0,\,k\mp 1,}{0}{m}-\Pnkm{0,\,k}{0}{m\pm 1},
\label{13-5.2}\\
[\Pnkm{k,\,0}{0}{m},\aocpm]=\delta_{k,\,0}\Pnkm{0,\,\mp 1}{0}{m}-\Pnkm{k,\,0}{0}{m\pm 1},\quad
[\Pnkm{0,\,l}{n}{0},\aocpm]=\Pnkm{0,\,l\mp 1}{n}{0}-\delta_{n,\,0}\Pnkm{0,\,l}{0}{\pm 1},\nn\end{gathered}$$ $$\label{13-5.3}
[\acopm,\aocp]= \Pnkm{0,\,1}{\mp 1}{0},\qquad  [\acopm,\aocm]=-\Pnkm{\pm 1,\,0}{0}{1}.$$ generate the ideal $I_0$ of the algebra $\widehat{\AG}_0$. Finally, we define the Lie algebra $\AG_0$ as the factor algebra $$\AG_0 = \widehat{\AG}_0 / I_0.$$ Thus algebra $\AG_0$ is the Lie algebra generated operators - satisfying the commutation relations -, -. This algebra is the closure of the associative algebra with generators satisfying the commutation relations -.

Algebra of generalized Chebyshev - Koornwinder oscillator
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In this section we construct and investigate the main object of our work — the algebra $\AG$ of two-dimensional generalized Chebyshev-Koornwinder oscillator that is a union of sectorial, radial, and boundary oscillators. In the work [@N46], the algebra $\AG$ was defined as a closure of the associative algebra $\widetilde{\AG}$ generated by operators $$\label{13-6.1}
\mathbb{I},\, \mathbb{N}_1,\, \mathbb{N}_2,\, \aspm,\, \arpm,\, \acopm,\, 