$$\label{13-4.11}
[\Pnkm{k,\,l}{k\pm t}{l\pm t},\arpm]=\sqrt{\frac{2}{5}}
\left(\Pnkm{k\mp 1,\,l\mp 1}{k\pm t}{l\pm t}-
\Pnkm{k,\,l}{k\pm t\pm 1}{l\pm t\pm 1}\right);$$

$$\label{13-4.12}
[\Pnkm{k,\,l}{k\pm t}{l\pm t},\Pnkm{m}{+}{n}]=
\Pnkm{k-n+m,\,m}{k\pm t}{n\pm t}-\Pnkm{k,\,m\mp t}{k\pm t-m+n}{n};$$

$$\label{13-4.13}
[\Pnkm{k,\,l}{k\pm t}{l\pm t},\Pnkm{m}{n}{+}]=
\Pnkm{m,\,l-n+m}{k\pm t}{l\pm t}-\Pnkm{m\mp t,\,l}{n}{l\pm t-m+n};$$

$$\label{13-4.14}
[\arm,\arp]=\frac{2}{5}\left(\Pnck{0}{0}-\Pnkc{0}{0}-\Pnkm{0,\,0}{0}{0}\right);$$

$$\label{13-4.15}
[\arpm,\Pnkm{m}{+}{n}]=\sqrt{\frac{2}{5}}\left(\Pnkm{m}{+}{n\pm 1}-\Pnkm{m\mp 1}{+}{n}\right);$$

$$\label{13-4.16}
[\arpm,\Pnkm{m}{n}{+}]=\sqrt{\frac{2}{5}}\left(\Pnkm{m}{n\pm 1}{+}-\Pnkm{m\mp 1}{n}{+}\right);$$

$$\label{13-4.17}
[\Pnkm{k}{+}{l},\Pnkm{m}{+}{n}]=\Pnkm{m}{+}{l}-\Pnkm{k}{+}{n}$$

$$\label{13-4.18}
[\Pnkm{k}{+}{l},\Pnkm{m}{n}{+}]=\Pnkm{m,\,k-n+m}{n-k+l}{l}-\Pnkm{m-l+k,\,k}{n}{l-m+n};$$

$$\label{13-4.19}
[\Pnkm{k}{l}{+},\Pnkm{m}{n}{+}]=\Pnkm{m}{l}{+}-\Pnkm{k}{n}{+}.$$

generate the ideal $I_r$ of the algebra $\AGh$. Finally, we determine the Lie algebra $\mathfrak{A}_r$ of radial oscillator as factor algebra $$\mathfrak{A}_r=\AGh/I_r.$$ Algebra $\mathfrak{A}_r$ is the Lie algebra with generators -, satisfying commutation relations -. The algebra $\mathfrak{A}_r$ is a closure of the associative algebra generated by the operators which satisfy the commutation relations .

Algebra of generalized boundary oscillator
==========================================

In this section we mainly use the same scheme of reasoning as in the previous two sections. Therefore, we give here only a sketch of our construction. We define the infinite-dimensional Lie algebra $\AG_0$ as the minimal extension of the associative algebra determined by the generators with the relations -. Thus $\AG_0$ is the closure of the boundary oscillator algebra.

We introduce the algebra $\widetilde{\AG}_0$, as an associative algebra which generators satisfy the commutation relations - . Then we define infinite-dimensional Lie algebra $\widehat{\AG}_0$ as algebra obtained from $\widetilde{\AG}_0$ by addition 