  In its customary formulation for one-component fluids, the Hierarchical
Reference Theory yields a quasilinear partial differential equation for an
auxiliary quantity f that can be solved even arbitrarily close to the critical
point, reproduces non-trivial scaling laws at the critical singularity, and
directly locates the binodal without the need for a Maxwell construction. In
the present contribution we present a systematic exploration of the possible
types of behavior of the PDE for thermodynamic states of diverging isothermal
compressibility kappa[T] as the renormalization group theoretical momentum
cutoff approaches zero. By purely analytical means we identify three classes of
asymptotic solutions compatible with infinite kappa[T], characterized by
uniform or slowly varying bounds on the curvature of f, by monotonicity of the
build-up of diverging kappa[T], and by stiffness of the PDE in part of its
domain, respectively. These scenarios are analzyed and discussed with respect
to their numerical properties. A seeming contradiction between two of these
alternatives and an asymptotic solution derived earlier [Parola et al., Phys.
Rev. E 48, 3321 (1993)] is easily resolved.
