  The B\"acklund transformations and the superposition formulas for two
sub--equations of the general elliptic equation are constructed from the
Riccati equation by using an indirect mapping method.The thirty-six previously
known solutions of the general elliptic equation are proved to be equivalent to
the another ten solutions.The classification of solutions for the general
elliptic equation is obtained based on the equivalence relations.The general
elliptic equation expansion method with new classified solutions are used to
obtain abundant new exact traveling wave solutions of a modified Camassa-Holm
equation.
