On solutions of the matrix nonlinear Schrödinger equation

Let $M_{nk}$ be the set of all complex $n\times k$-matrices. Consider the equation $iu_t=u_{xx}+uAu^*Bu$ for an unknown $M_{nk}$-valued function $u(x,t)$ of two real variables $x,t$, where $A\in M_{kk}$ and $B\in M_{nn}$ are non-degenerate Hermitian matrices and the star stands for Hermitian conjugation. We prove that every real analytic solution is a globally meromorphic function of $x$ for every fixed $t$. When all the eigenvalues of both matrices $A,B$ are of the same sign, every local real analytic solution extends to a real analytic function in a strip (depending on the solution) parallel to the $x$-axis (possibly a half-plane or the whole plane), and every such strip carries a solution inextensible beyond it.