Formally integrable Mizohata systems of codimension 1

In the paper we prove that any formally integrable Mizohata system of codimension one
$$\left \{
\begin{array}{@{}l@{}} \partial_1u=\epsilon_1ix^1\partial_nu+f_1, \\ \partial_2u=\epsilon_2ix^2\partial_nu+f_2, \\ \dots \dots \dots \\ \partial_{n-1}u=\epsilon_{n-1}ix^{n-1}\partial_nu+f_{n-1} \end{array}
\right. $$
can be reduced by a local change of the variables to a system of the form
$$\left \{
\begin{array}{@{}l@{}} \partial_1v^1+\partial_2v^2=\psi _1, \\ \partial_1v^2-\partial_2v^1=\psi _2 \end{array}
\right. $$
and, consequently, to Poisson's equation in the plane.