  A N=4 supersymmetric matrix KP hierarchy is proposed and a wide class of its
reductions which are characterized by a finite number of fields are described.
This class includes the one-dimensional reduction of the two-dimensional
N=(2|2) superconformal Toda lattice hierarchy possessing the N=4 supersymmetry
-- the N=4 Toda chain hierarchy -- which may be relevant in the construction of
supersymmetric matrix models. The Lax pair representations of the bosonic and
fermionic flows, corresponding local and nonlocal Hamiltonians, finite and
infinite discrete symmetries, the first two Hamiltonian structures and the
recursion operator connecting all evolution equations and the Hamiltonian
structures of the N=4 Toda chain hierarchy are constructed in explicit form.
Its secondary reduction to the N=2 supersymmetric alpha=-2 KdV hierarchy is
discussed.
