  In the present paper we continue our reconsideration about the foundations
for a thermostatistical description of the called Hamiltonian nonextensive
systems (see in cond-mat/0604290). After reviewing the selfsimilarity concept
and the necessary conditions for the ensemble equivalence, we introduce the
reparametrization invariance of the microcanonical description as an internal
symmetry associated with the dynamical origin of this ensemble. Possibility of
developing a geometrical formulation of the thermodynamic formalism based on
this symmetry is discussed, with a consequent revision about the classification
of phase-transitions based on the concavity of the Boltzmann entropy. The
relevance of such conceptions are analyzed by considering the called Antonov
isothermal model.
