Integrable and nonintegrable cases of the Lax equations with a source

The Korteweg–de Vries equation with a source given as a Fourier integral over eigenfunctions of the so-called generating operator is considered. It is shown that depending on the choice of a basis of eigenfunctions we have the following three possibilities: 1) evolution equations for the scattering data are nonintegrable; 2) evolution equations for the scattering data are integrable but the solution of the Cauchy problem for the Korteweg–de Vries equation with a source at some $t'>t_0$ leaves the considered class of functions decreasing rapidly enough as $x\to \pm \infty$; 3) evolution equations for the scattering data are integrable and the solution of the Cauchy problem for the Korteweg–de Vries equation with a source exists at all $t>t_0$. All these possibilities are widespread and occur in other Lax equations with a source.