Painlevé analysis and integrability of two-coupled non-linear oscillators

Integrability of a linearly damped two-coupled non-linear oscillators equation\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{gathered} \mathop x\limits^{..} = - d\mathop {\mathop x\limits^. - \alpha x - \delta _1 (x^2 + y^2 ) - 2\delta _2 xy}\limits^. \hfill \\ \mathop y\limits^{..} = d\mathop y\limits^. - \beta y - \delta _2 (x^2 + y^2 ) - 2\delta _1 xy \hfill \\ \end{gathered} $$ \end{document} is investigated by employing the Painlevé analysis. The following two integrable cases are identified: (i)d = 0, α =β, δ_1 and δ_2 are arbitrary, (ii) d^2= 25α/6, α =β, δ_1 and δ_2 are arbitrary. Exact analytical solution is constructed for the integrable choices.