Аннотация:
In this work, the questions of integrability through a finite set
of elementary functions of nonconservative dynamical systems of the
following form
$$
\dot{\alpha}=f_{\alpha}(\omega,\sin\alpha,\cos\alpha),
\quad
\dot{\omega}_{k}=f_{k}(\omega,\sin\alpha,\cos\alpha),
\quad
k=1,\ldots,n
$$
are studied.
The system is given on $S^{1}\{\alpha\bmod2\pi\}\times
\mathbb{R}^{n}\{\omega\}$, $\omega
=(\omega_{1},\ldots,\omega_{n})$, where the functions
$f_{\lambda}(\omega,\sin\alpha,\cos\alpha)$,
${\lambda=\alpha,1,\ldots,n}$ are the following ones:
\begin{gather*}
f_{\alpha}(-t_{1},-t_{2},t_{3})=-f_{\alpha}(t_{1},t_{2},t_{3}),
\quad
f_{\alpha}(t_{1},t_{2},-t_{3})=f_{\alpha}(t_{1},t_{2},t_{3}),
\\
f_{k}(-t_{1},-t_{2},t_{3})=-f_{k}(t_{1},t_{2},t_{3}),
\quad
f_{k}(t_{1},t_{2},-t_{3})=-f_{k}(t_{1},t_{2},t_{3}).
\end{gather*}
Such a system corresponds to the system
$$
\frac{d\omega_{k}}{d\alpha}=
\frac{f_{k}(\omega,\sin\alpha,\cos\alpha)}
{f_{\alpha}(\omega,\sin\alpha,\cos\alpha)},
$$
which, by using the substitution $\tau=\sin\alpha$, is reduced to the form
$$
\frac{d\omega_{k}}{d\tau}=
\frac{f_{k}(\omega,\tau,\varphi_{k}(\tau))}
{f_{\alpha}(\omega,\tau,\varphi_{\alpha}(\tau))},
$$
where $\varphi_{\lambda}(-\tau)=\varphi_{\lambda}(\tau)$,
$\lambda=\alpha,1,\ldots,n$.
In the work, the case where the functions
$f_{\lambda}(\cdot,\cdot,\cdot)$, $\lambda=\alpha,1,\ldots,n$,
are polynomials in $\omega$, $\tau$ is considered.
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