  In this paper, Liouville soliton surfaces based on some soliton solutions of
the Liouville equation are constructed and displayed graphically, including
some of those corresponding to Darboux-transformed counterparts. We find that
the Liouville soliton surfaces are centroaffine surfaces of Tzitzeica type and
their centroaffine invariant can be expressed in terms of the Hamiltonian. The
traveling wave solutions to Liouville equation from which these soliton
surfaces stem are also obtained through a modified variation of parameters
method which is shown to lead to elliptic functions solution method.
