  We propose an extension of the Dubrovin-Zhang perturbative approach to the
study of normal forms for non-Hamiltonian integrable scalar conservation laws.
The explicit computation of the first few corrections leads to the conjecture
that such normal forms are parameterized by one single functional parameter,
named viscous central invariant. A constant valued viscous central invariant
corresponds to the well-known Burgers hierarchy. The case of a linear viscous
central invariant provides a viscous analog of the Camassa-Holm equation, that
formerly appeared as a reduction of a two-component Hamiltonian integrable
systems. We write explicitly the negative and positive hierarchy associated
with this equation and prove the integrability showing that they can be mapped
respectively into the heat hierarchy and its negative counterpart, named the
Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial
data is also proven. We show how transport equations can be used to effectively
construct asymptotic solutions via an extension of the quasi-Miura map that
preserves the initial datum. The method is alternative to the method of the
string equation for Hamiltonian conservation laws and naturally extends to the
viscous case. Using these tools we derive the viscous analog of the Painlev\'e
I2 equation that describes the universal behaviour of the solution at the
critical point of gradient catastrophe.
