Integration of the Korteweg–de Vries type equations with a loaded term in the class of periodic functions

In this paper, we study the integrability of a Korteweg–de Vries type equation with a loaded term in the class of periodic functions using direct and inverse spectral problems posed for the Sturm–Liouville operator on the entire axis. We present some information about the Sturm–Liouville operator with a periodic coefficient and its application to solving a loaded Korteweg–de Vries type equation. We show that the function constructed using the obtained system of Dubrovin differential equations and trace formulas is a solution to the problem posed. We present important consequences about the period of the solution with respect to $x$ and about the analyticity of the solution with respect to $x$ for a Korteweg–de Vries type equation with a loaded term.