  I prove the recently conjectured relation between the $2\times 2$-matrix
differential operator $L=\partial^2-U$, and a certain non-linear and non-local
Poisson bracket algebra ($V$-algebra), containing a Virasoro subalgebra, which
appeared in the study of a non-abelian Toda field theory. Here, I show that
this $V$-algebra is precisely given by the second Gelfand-Dikii bracket
associated with $L$. The Miura transformation is given which relates the second
to the first Gelfand-Dikii bracket. The two Gelfand-Dikii brackets are also
obtained from the associated (integro-) differential equation satisfied by
fermion bilinears. The asymptotic expansion of the resolvent of $(L-\xi)\Psi=0$
is studied and its coefficients $R_l$ yield an infinite sequence of
hamiltonians with mutually vanishing Poisson brackets. I recall how this leads
to a matrix KdV hierarchy which are flow equations for the three component
fields $T, V^+, V^-$ of $U$. For $V^\pm=0$ they reduce to the ordinary KdV
hierarchy. The corresponding matrix mKdV equations are also given, as well as
the relation to the pseudo- differential operator approach. Most of the results
continue to hold if $U$ is a hermitian $n\times n$-matrix. Conjectures are made
about $n\times n$-matrix $m^{\rm th}$-order differential operators $L$ and
associated $V_{(n,m)}$-algebras.
