  The Dubrovin-Zhang hierarchy is a Hamiltonian infinite-dimensional integrable
system associated to a semi-simple cohomological field theory or,
alternatively, to a semi-simple Dubrovin-Frobenius manifold. Under an extra
assumption of homogeneity, Dubrovin and Zhang conjectured that there exists a
second Poisson bracket that endows their hierarchy with a bi-Hamiltonian
structure. More precisely, they gave a construction for the second bracket, but
the polynomiality of its coefficients in the dispersion parameter expansion is
yet to be proved. In this paper we use the bi-Hamiltonian recursion and a set
of relations in the tautological rings of the moduli spaces of curves derived
by Liu and Pandharipande in order to analyze the second Poisson bracket of
Dubrovin and Zhang. We give a new proof of a theorem of Dubrovin and Zhang that
the coefficients of the dispersion parameter expansion of the second bracket
are rational functions with prescribed singularities. We also prove that all
terms in the expansion of the second bracket in the dispersion parameter that
cannot be realized by polynomials because they have negative degree do vanish,
thus partly confirming the conjecture of Dubrovin and Zhang.
