sectorial oscillator, the Hamiltonian $H_r$ radial oscillator contains an extra term $I_B\left(\ts\frac{1}{8};\ts\frac{1}{2}\right)$ in addition to standard members.

In the work [@N46] the algebra $\mathfrak{A}_r$ of generalized radial oscillator was defined as a closure of the associative algebra generated by operators $$\label{13-2.29}
a^{\pm}_{rad},\quad \mathbb{N}_1,\quad \mathbb{N}_2,\quad I_B(\ts\half,\ts\half),
\quad \mathbb{I},$$ satisfying the following commutation relations $$\begin{gathered}
[a^{-}_{rad},a^{+}_{rad}]=\frac{4}{5}I_B(\ts\half,\ts\half);
\quad a^{-}_{rad}I_B(\ts\half,\ts\half)=0;\quad I_B(\ts\half,\ts\half)a^{+}_{rad}=0;\nn\\
[\mathbb{N}_1,a^{\pm}_{rad}]=\pm a^{\pm}_{rad};\quad
[\mathbb{N}_2,a^{\pm}_{rad}]=\pm a^{\pm}_{rad};\quad
[\mathbb{N}_1,I_B(\ts\half,\ts\half)]=0,\quad
[\mathbb{N}_2,I_B(\ts\half,\ts\half)]=0. \label{13-2.30}\end{gathered}$$

### Generalized boundary oscillator 

Recall that $\Hco$ and $\Hoc$ denote the closures in $\H$ of linear spans of the sets $\bask$ and $\basl$. To define the boundary oscillator we use [@N46] the decomposition of the Hilbert spaces $\Hco$ and $\Hoc$ $$\label{13-2.31}
\Hco=\bigoplus_{N=0}^{\infty}\left(\H^{(N)}\cap\Hco\right)\quad
\Hoc=\bigoplus_{N=0}^{\infty}\left(\H^{(N)}\cap\Hoc\right)$$ and operators $P^{m,\,n}_{s,\,t}\, \suchthat\, P^{m,\,n}_{s,\,t}U_{k,\,l}=
\delta_{k,\,m}\delta_{l,\,n}U_{s,\,t}$. The ladder operators are defined by decompositions (see Fig.2) $$\label{13-2.32}
\acopm=\bigoplus_{m=0}^{\infty}P_{m\pm 1,\,0}^{m,\,0},\quad
\aocpm=\bigoplus_{m=0}^{\infty}P_{0,\,m\pm 1}^{0,\,m},$$ and have [@N47] the following differential operator representation: $$\acopm\rule[-10pt]{.2pt}{20pt}_{\,\,\H^{(N)}}
=U_{N\pm 1,\,0}\frac{\mathbf{D}_{N,\,0}}{N!};\quad
\aocpm\rule[-10pt]{.2pt}{20pt}_{\,\,\H^{(N)}}
=U_{0,\,N\pm 1}\frac{\mathbf{D}_{0,\,N}}{N!}.$$ For position operators $$\begin{aligned}
\acop+\acom&=\left(Z+\overline{Z}-a_{sect}s^+-a_{sect}^+Z+
(a_{sect}^+)^2\right)\mathbb{P}_1,\nn \\
\aocp+\aocm&=\left(Z+\overline{Z}-a_{sect}^--a_{sect}^-Z+
(a_{sect}^-)^2\right)\mathbb{P}_2,\end{aligned}$$ this gives [@N47] the following representation by differential operators: $$\begin{gathered}
\acop+\acom=Z+\overline{Z}-\frac{1}{\sqrt{10}} \times
\left\{ \bigoplus_{N=1}^{\infty} \left[\left(
\sum_{m=0}^N \frac{U_{m-1,\,N-m+1}\,\mathbf{D}_{m,\,N-m}}{m!(N-m)!}
\right)(\mathbb{I}+Z)\right.\right.
\\
-\left.\left.
\left(\sum_{m=0}^{N-2}
\frac{U_{m,\,N-m}\,\mathbf{D}_{m+2,\,N-m-2}}{(m+2)!(N-m-2)!}\right)
\right]\right\}\mathbb{P}_1;\end{gathered}$$ $$\begin{gathered}
\aocp+\aocm=Z+\overline{Z}-\frac{1}{\sqrt{10}} \times
\left\{\bigoplus_{N=1}^{\infty}\left[\left(
\sum_{m=0}^N \frac{U_{m+1,\,N-m-1}\,\mathbf{D}_{m,\,N-m}}{m!(N-m)!}
\right)(\mathbb{I}+\overline{Z})\right.\right. \\
-\left.\left.\left(\sum_{m=0}^{N-2}
\frac{U_{m+2,\,N-m-2}\,\mathbf{D}_{m,\,N-m}}{m!(N-m)!}\right)
\right]\right\}\mathbb{P}_2.\end{gathered}$$ The quadratic Hamiltonian of the boundary oscillator has the form $$H_0=\frac15\left(H_{\bullet,\,0}+H_{0,\,\bullet}-P^{0,\,0}_{0,\,0}\right),$$ where $$\begin{aligned}
H_{\bullet,\,0}=
\left(a^{+}_{\bullet,\,0}a^{-}_{\bullet,\,0}+a^{-}_{\bullet,\,0}a^{+}_{\bullet,\,0}\right),\\
H_{0,\,\bullet}=
