  In 1974 P. Lax introduced an algebro-analytic mechanism similar to the Lax
L-A pair. Using it we prove global existence and uniqueness for solutions of
the initial value problem for mKdV in classes of smooth functions which can be
unbounded at infinity, and may even include functions which tend to infinity
with respect to the space variable. Moreover, we establish the invariance of
the spectrum and the unitary type of the Schr{\"o}dinger operator under the KdV
flow and the invariance of the spectrum and the unitary type of the impedance
operator under the mKdV flow for potentials in these classes.
