Motivation: I'm trying to understand the Section 4 in the this section 4 in paper" Low regularity conservation laws for integrable PDE by Killip-Visan-Zhang ö

It reads as follows: many completely integrable PDE admit a Lax pair of the following form: $$\frac{d}{dt} L(t,\kappa)=[P(t, \kappa), L(t, \kappa)]$$ with $$L(t,\kappa)= \begin{bmatrix} -\partial+ \kappa & iq(x)\\ \mp I \bar{q}(x) & -\partial-\kappa \end{bmatrix} $$

and some operator pencil $P(t, \kappa).$

My Question: Can we say following cubic NLS $$-i\frac{d}{dt}q= -\beta(t)q'' \pm 2|q|^2q$$ where $\beta(t+1)=\beta(t)$ for all $t\in \mathbb R$ and $\beta$ is piecewise constant with finitely many discontinuity in $[0,1],$ admit aLax pair? If so what is $L$ and $P$?