Spectral theory for periodic vector NLS equations

We consider a first order operator with a periodic 3x3 matrix potential on the real line. This operator appears in the problem of the periodic vector NLS equation. The spectrum of the operator covers the real line, it is union of the spectral bands of multiplicity 3, separated by intervals (gaps) of multiplicity 1. The main results of this work are the following:
1) The Lyapunov function on the corresponding 2 or 3-sheeted Riemann surface is described.
2) Necessary and sufficient conditions are given when the Riemann surface is 2-sheeted.
3) The asymptotics of 2-periodic eigenvalues are determined.
4) One constructs an entire function, which is positive on the spectrum of multiplicity 3 and is negative on its gaps.
5) The estimate of the potential in terms of gap lengths is obtained.
6) The Borg type results about inverse problems are solved.
7) The solution of the periodic vector NLS equation for the case of the 2-sheeted Riemann surface is described.