  In this paper we prove an algebraic criterion which characterizes singular
quasi-bi-hamiltonian structures constructed on the lines of a general, simple,
new formal procedure proposed by the authors. This procedure shows that for the
definition of a quasi-bi-hamiltonian system the requirement of non-singular
Poisson tensors, contained in the original definition by Brouzet et al., is not
essential. Besides, it is incidentally shown that one method of constructing
Poisson tensors available in the literature is a particular case of ours. We
present 2 examples.
