  We consider the generalized matrix non-linear Schrodinger (NLS) hierarchy. By
employing the universal Darboux-dressing scheme we derive solutions for the
hierarchy of integrable PDEs via solutions of the matrix
Gelfand-Levitan-Marchenko equation, and we also identify recursion relations
that yield the Lax pairs for the whole matrix NLS-type hierarchy. These results
are obtained considering either matrix-integral or general $n^{th}$ order
matrix-differential operators as Darboux-dressing transformations. In this
framework special links with the Airy and Burgers equations are also discussed.
The matrix version of the Darboux transform is also examined leading to the
non-commutative version of the Riccati equation. The non-commutative Riccati
equation is solved and hence suitable conserved quantities are derived. In this
context we also discuss the infinite dimensional case of the NLS matrix model
as it provides a suitable candidate for a quantum version of the usual NLS
model. Similarly, the non-commutitave Riccati equation for the general dressing
transform is derived and it is naturally equivalent to the one emerging from
the solution of the auxiliary linear problem.
