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We consider perturbations of the special pole-free joint solution $U(x,t)$ ofthe 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 theKdV 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 theRiemann-Hilbert problem method. Namely, we construct the corresponding spectralfunctions $a(\lambda), r(\lambda),$ and give characterization of the compactperturbations in terms of $a(\lambda), r(\lambda).$

On a class of compact perturbations of the special pole-free joint solution of KdV and $P_I^2.$