Trace formulas for the one-dimensional Stark operator and integrals of motion for the cylindrical Korteweg-de Vries equation

A complete series of trace formulas for the one-dimensional Stark operator on the entire axis with a rapidly decreasing potential is constructed. The Stark equation is related to the cylindrical Korteweg-de Vries equation by the pair ( $ L$–$ A$). For this equation, infinite series of integrals of motion is constructed that corresponds to the trace formulas for the Stark operator.