  We prove that linear degeneracy is a necessary conditions for systems in
Jordan-block form to admit a compatible quasilinear differential constraint.
Such condition is also sufficient for 2x2 systems and turns out to be
equivalent to possess the Hamiltonian property. Some explicit solutions of
parabolic systems are herein given: two principal hierarchies arising from the
associativity theory and the delta-functional reduction of the El's equation in
the hard rod case are integrated.
