  The string equation appearing in the double scaling limit of the Hermitian
one--matrix model, which corresponds to a Galilean self--similar condition for
the KdV hierarchy, is reformulated as a scaling self--similar condition for the
Ur--KdV hierarchy. A non--scaling limit analysis of the one--matrix model has
led to the complexified NLS hierarchy and a string equation. We show that this
corresponds to the Galilean self--similarity condition for the AKNS hierarchy
and also its equivalence to a scaling self--similar condition for the
Heisenberg ferromagnet hierarchy.
