of $H_s$ with eigenvalues $$\lambda_{0,0}=0;\, \lambda_{N,0}=\lambda_{0,N}=\frac{1}{10}, \,(N\geq1);\,
\lambda_{k,\,l}=\frac{1}{5},\, (k,l\geq1).$$ We consider $\H$ as a Fock space and define ladder operators $\aspm$ by the relations $$\asp=\frac{1}{\sqrt{40}}\left(Z+Z^{\dag}\right),\quad
\asm=\frac{1}{\sqrt{40}}\left(\overline{Z}+\overline{Z}^{\dag}\right).$$ These operators acting in “$N$-particle” subspaces $\H^{(N)}$ ($N=k+l$) as follows $$\label{13-2.10}
\aspm U_{k,\,l}(z,\,\z)=\frac{1}{\sqrt{10}}U_{k\mp 1,\,l\pm 1}(z,\,\z),\qquad
\asp U_{0,N}(z,\,\z)=0,\quad \asm U_{N,0}(z,\,\z)=0.$$ Using , we have shown in [@N47] that the ladder operators can be represented by the differential operators: $$\aspm=\frac{1}{\sqrt{10}}\bigoplus_{N=1}^{\infty}\sum_{m=0}^{N}
U_{m\mp 1,\,N-(m\mp 1)}\frac{\mathbf{D}_{m,\,N-m}}{m!(N-m)!},$$ where $$\mathbf{D}_{m,\,N-m}=\frac{\partial^N}{\partial z^m \partial \z^{N-m}}.$$ The ladder operators are adjoint to each other $\left(\aspm)\right)^*=\asmp$, and Hamiltonian in terms of these operators has the form $$H_s=H_s^{(1)}+H_s^{(2)},\qquad
H_s^{(1)}=a_{sect}^{+}a_{sect}^{-},\quad H_s^{(2)}=a_{sect}^{-}a_{sect}^{+}.$$ Self-adjoint “number” operators $\mathbb{N}_1$, $\mathbb{N}_2$ are defined by their action on the basic elements of the space $\H$: $$\mathbb{N}_1U_{k,\,l}=kU_{k,\,l},\quad
\mathbb{N}_2U_{k,\,l}=lU_{k,\,l}.$$ We also need two auxiliary operators $$\label{13-2.15}
\mathbb{P}_1=\bigoplus_{N=0}^{\infty}P_{N,\,0},\qquad \mathbb{P}_2=
\bigoplus_{N=0}^{\infty}P_{0,\,N},$$ where $$P_{N,\,0}U_{k,\,l}(z,\,\z)=\delta_{k,\,N}\delta_{l,\,0}U_{k,\,l}(z,\,\z),\quad
P_{0,\,N}U_{k,\,l}(z,\,\z)=\delta_{k,\,0}\delta_{l,\,N}U_{k,\,l}(z,\,\z).$$

In [@N46] the algebra $\mathfrak{A}_s$ of generalized sectorial oscillator was defined as the closure of an associative algebra generated by operators $$\label{13-2.16}
\mathbb{I},\, a_{sect}^{\pm},\, \mathbb{N}_1,\, \mathbb{N}_2,\, \mathbb{P}_1,\, \mathbb{P}_2,$$ satisfying the commutation relations $$\begin{gathered}
[a_{sect}^{+},a_{sect}^{-}]=\frac{1}{10}\left(\mathbb{P}_1-\mathbb{P}_2\right);\quad
[\mathbb{N}_1,a_{sect}^{\pm}]=\mp a_{sect}^{\pm};\quad
[\mathbb{N}_2,a_{sect}^{\pm}]=\pm a_{sect}^{\pm};\nn\\
[\mathbb{P}_1,\mathbb{P}_2]=0,\quad [\mathbb{N}_1,\mathbb{N}_2]=0,\quad
[\mathbb{N}_i,\mathbb{P}_j]=0,\,(i,j=1,2);\label{13-2.17}\\
\mathbb{P}_1a_{sect}^{+}=0,\quad a_{sect}^{-}\mathbb{P}_1=0,\quad
a_{sect}^{+}\mathbb{P}_2=0,\quad \mathbb{P}_2a_{sect}^{-}=0.\nn\end{gathered}$$

### Generalized radial oscillator 

As in the case of sectorial oscillator, we start by defining the “position” operator $$\label{13-2.18}
X:=\!-5H_s\!-\!\frac{1}{4}\left(Z\Z^{\,*}+Z^*\Z+\Z Z^*+\Z^{\,*}Z\right),\quad \text{Dom}[X]\!=\!\H.$$ By analogy with , using the 