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This study investigates the Slepyan–Palmov (SP) model, which describes plane longitudinal waves propagating within a medium comprising a carrier medium and nonlinear oscillators. The primary objective is to analyze the integrability properties of this model. The research entails two key aspects. Firstly, the study explores the group invariant solution by utilizing reductions in symmetry subalgebras based on the optimal system. Secondly, the conservation laws are studied using the homotopy operator, which offers advantages over the conventional multiplier approach, especially when arbitrary functions are absent from both the equation and characteristics. This method proves advantageous in handling complex multipliers and yields significant outcomes.