Conjugate problem for third order equation with multiple characteristics and positive function on the highest derivative

The conjugate problem (generalized diffraction problem) is studied for third order equation $u_{t}-h(x)u_{xxx}+c(x,t)u=f(x,t)$, where coefficient $h(x)$ is positive and may have a discontinuity of the first kind at the point $x=0$. The existence and uniqueness of regular solutions are established.