On vector derivative nonlinear Schrödinger equation

We propose a sequence of Lax pairs, the compatibility conditions of which are integrable vector nonlinear equations. The first equations in this hierarchy are vector Kaup — Newell, Chen — Lee — Liu, Gerdjikov — Ivanov integrable nonlinear equations. The type of vector equation depends on an additional parameter $\alpha$. The proposed form of the vector Kaup — Newell equation has slight differences in comparison with the classical form. We show that the evolution of simplest nontrivial solutions of these equations is a composition of the evolutions of length and orientations of solution. We study properties of spectral curves of simplest nontrivial solutions the vector equations in the constructed hierarchy.