The inverse scattering problem for the perturbed string equation in characteristic variables on the whole axis is studied. Using the generalized Lax equation generated by the perturbed string equation, we derive the time-evolution of the scattering operator and the two-dimensional generalization from the one-dimensional Korteweg-de Vries equation. This enables us to solve the system of time-dependent fundamental equations in the inverse problem. Then, the solution of the two-dimensional generalization from the Korteweg-de Vries equation is found that is expressed through the found solution of this system.