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## Liouvillian Integrability and the Poincaré Problem for Nonlinear Oscillators with Quadratic Damping and Polynomial Forces

Journal of Dynamical and Control Systems. 2021. Vol. 27. No. 2. P. 403-415.
Demina M.V., Kuznetsov N.S.

The upper bound on the degrees of irreducible Darboux polynomials associated
to the ordinary differential equations $x_{tt}+\varepsilon {x_t}^2+\eta x_t+f(x)=0$ with $f(x)\in\mathbb{C}[x]\setminus\mathbb{C}$ and $\varepsilon\neq0$ is derived. The availability of this bound provides the
solution of the Poincar\'{e} problem. Results on uniqueness and existence of
Darboux polynomials are presented. The problem of Liouvillian integrability
for related dynamical systems is solved completely. It is proved that
Liouvillian first integrals exist if and only if $\eta=0$.