The Dirac operator with elliptic potential

The Dirac operator with elliptic finite-gap potential
$$ -\mathrm i\begin{pmatrix}1&0\\0&-1\end{pmatrix}\Psi _x +\begin{pmatrix}0&p\\q&0\end{pmatrix}\Psi =\lambda\Psi . $$
is considered. An Ansatz for the Krichever curves associated with elliptic (in $x$) finite-gap solutions of the 'decomposed' non-linear Schrödinger equation
$$ \begin{cases} \mathrm ip_t+p_{xx}-2p^2q=0, \\iq_t-q_{xx}+2pq^2=0 \end{cases} $$
and of the modified $KdV$ ($mKdV$) equation
$$ \begin{cases} p_t+p_{xxx}-6pqp_x=0, \\q_t+q_{xxx}-6pqq_x=0. \end{cases} $$
is presented. Examples of two- and three-sheeted coverings associated with the one- and twogap Dirac potential are discussed.