  We show that the 2-matrix string model corresponds to a coupled system of
$2+1$-dimensional KP and modified KP ($\KPm$) integrable equations subject to a
specific ``symmetry'' constraint. The latter together with the
Miura-Konopelchenko map for $\KPm$ are the continuum incarnation of the matrix
string equation. The $\KPm$ Miura and B\"{a}cklund transformations are natural
consequences of the underlying lattice structure. The constrained $\KPm$ system
is equivalent to a $1+1$-dimensional generalized KP-KdV hierarchy related to
graded ${\bf SL(3,1)}$. We provide an explicit representation of this
hierarchy, including the associated ${\bf W(2,1)}$-algebra of the second
Hamiltonian structure, in terms of free currents.
