  We construct the classical W-algebras for some non-abelian Toda systems
associated with the Lie groups GL(2n,R) and Sp(n,R). We start with the set of
characteristic integrals and find the Poisson brackets for the corresponding
Hamiltonian counterparts. The convenient block matrix representation for the
Toda equations is used. The infinitesimal symmetry transformations generated by
the elements of the W-algebras are presented.
