1\right) }(x,t)$ is meromorphic in $x$ and $\mathbb{M}^{\left( 2\right) }(x,t)$ is Gevrey $G^{\frac{1}{2\varepsilon }-1}$ regular. If in addition $1+\mathbb{M}(x,t)$ is invertible for any real $x$ and $t>0$ then the problem has a solution $u(x,t)$ given by $$u(x,t)=-2\partial _{x}^{2}\log \det \left( 1+\mathbb{M}(x,t)\right) ,
\label{det_form1}$$belonging to the Gevrey class $G_{\limfunc{loc}}^{\frac{1}{2\varepsilon }-1}$.

Theorems \[thm1.2\] and \[thm1.3\] significantly improve our results in [@Ry11] which in turn improve Tarama [@Tarama04]. Theorems [thm1.2]{} and \[thm1.3\] have also some important corollaries. We will come back to the relevant discussions in the last section when we have the necessary background. We only mention here that our approach is based on the Inverse Scattering Transform (IST) combined with pseudo-analytic continuation techniques developed by E.M. Dyn’kin (see e.g. [@Dyn76], [@BorDyn93]) and we do not believe that any of the statements of Theorem \[thm1.2\] can be obtained by purely PDE techniques.

The paper is organized as follows. In Section 2, for the reader’s convenience we list our main notation and give the relevant preliminaries. In Section 3 we define a suitable reflection coefficient and investigate its properties which will play a central role in our consideration. The results of this section may have some independent interest. In Section 4 we give 