The paper investigates the generalized Lazer-Solimini periodic equation \[ x'' + g(x) = p(t), \] where \(g\) is a locally Lipschitz continuous function, positive and non-increasing on \((0,+\infty)\), with \(\lim_{x\to0^{+}} g(x)=\infty\) and \(\int_{0}^{1} g(x)\,\mathrm{d}x<+\infty\), \(p\) is a continuous function, \(2\pi\)-periodic on \(\mathbb{R}\), such that \(\int_{0}^{2\pi} p(s)\,\mathrm{d}s / 2\pi > \lim_{x\to+\infty} g(x)\). Exploiting a generalized version of the Poincaré-Birkhoff fixed point theorem, the authors prove the existence of a constant \(K>0\) such that there exist at least two \(2\pi\)-periodic bouncing solutions (reaching the singularity at isolated points) with maximum less than \(K\) and a unique \(2\pi\)-periodic classical solution with minimum greater than \(K\).