Integrabilities and solutions of a perturbed Benjamin-Ono equation

In this paper, we investigate the integrability and solutions of a perturbed Benjamin-Ono equation, which arises from an integrable classification based on a perturbative symmetry approach. First, we provide a classification of general Benjamin-Ono type systems that admit solutions related to the Calogero-Moser system, including the perturbed Benjamin-Ono equation. Second, we demonstrate that this equation possesses a Lax pair. Third, we show that the equation can be transformed into a linear form through an appropriate change of the dependent variable. Finally, we derive both non-periodic and periodic solutions under suitable initial conditions.