Quasi-periodic solutions of a damped reversible oscillator at resonance

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We prove the existence of quasi-periodic solutions and of Aubry-Mather sets for a resonant reversible equation of the form $x^{\prime\prime} + ax^+ - bx^- +\varphi(x) +f(x,x',t)= p(t)$; the functions $p$ and $f$ are $2\pi$-periodic in $t$ and the perturbation $\varphi$ is bounded.