  We consider perturbations of the special pole-free joint solution $U(x,t)$ of
the Korteweg--de Vries equation $u_t+uu_x+\frac{1}{12}u_{xxx}=0$ and $P_I^2$
equation $u_{xxxx}+10u_x^2+20uu_{xx}+40(u^3-6tu+6x)=0$ under the action of the
KdV flow. We show that if the perturbation is compact and of bounded variation,
then the initial value problem for the KdV equation has a classical solution.
Our method is the inverse scattering transform method in the form of the
Riemann-Hilbert problem method. Namely, we construct the corresponding spectral
functions $a(\lambda), r(\lambda),$ and give characterization of the compact
perturbations in terms of $a(\lambda), r(\lambda).$
