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 ε ≠ 0 is derived. The availability of this bound provides the solution of the Poincaré 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 η = 0.

中文翻译：

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具有二次阻尼和多项式力的非线性振荡器的Liouvillian可积性和Poincaré问题

与常微分方程\（x_ {tt} + \ varepsilon {x_ {t}} ^ {2} + \ eta x_ {t} + f（x）= 0 \不可约的Darboux多项式的阶数的上限），其中\（f（x）\ in \ mathbb {C} [x] \ setminus \ mathbb {C} \），并且ε ≠0。此界限的可用性提供了庞加莱问题的解决方案。给出了Darboux多项式唯一性和存在性的结果。完全解决了相关动力学系统的Liouvillian可积性问题。证明当且仅当η = 0时，存在Liouvillian第一积分。