  In this work we consider integrable PDE with higher dimensional Lax pairs.
Our main example is a quadratic dNLS equation with a $3 \times 3$ Lax pair. For
this equation we show a-priori estimates in Sobolev spaces of negative
regularity $H^s(\mathbb{R}), s > -\frac{1}{2}$. We also prove that for general
$N \times N$ Lax operators $L$, the transmission coefficient coincides with the
$2$-renormalized perturbation determinant.
