  In this paper, the authors investigate non-homogeneous Hamiltonian operators
composed of a first-order Dubrovin-Novikov operator and an ultralocal one. The
study of such operators turns out to be fundamental for the inverted system of
equations associated with a class of Hamiltonian scalar equations. Often, the
involved operators are degenerate in the first-order term. For this reason, a
complete classification of the operators with degenerate leading coefficient in
systems of 2 and 3 components is presented.
