The work is a continuation of the research started in the author’s previous works. The purpose of this work is to study some properties of a nonlinear system of partial differential equations, which is a perturbation of the Korteweg–de Vries equation. Studies on scale invariance and the Hirota method are used. Based on the obtained invariants, two types of self-similar solutions are constructed in the form of power series. The Hirota method proved the existence of a 1-soliton and 2-soliton solution. It is shown that if the kink is chosen as the perturbation, the perturbed Korteveg-de Vries equation preserves the solution in the form of a solitary wave. It is shown that the system describes the interactions of a soliton and a kink, a soliton and a soliton on a “pedestal”, two solitons on a “pedestal”.