  The purpose of this paper is to develop the negative order MKdV hierarchy and
to present a new related integrable Neumann-like Hamiltonian flow from the view
point of inverse recursion operator and constraint method. The whole MKdV
hierarchy both positive and negative is generated by the kernel elements of
Lenard's operators pair and recursion operator. Through solving a key operator
equation, the whole MKdV hierarchy is shown to have the Lax representation. In
particular, some new integrable equation together with the Liouville equations,
the sine-Gordon equation, and the sinh-Gordon equation are derived from the
negative order MKdV hierarchy. It is very interesting that the restricted flow,
corresponding to the negative order MKdV hierarchy, is just a new kind of
Neumann-like system. This new Neumann-like system is obtained through
restricting the MKdV spectral problem onto a symplectic submanifold and is
proven to be completely integrable under the Dirac-Poisson bracket, which we
define on the symplectic submanifold. Finally, with the help of the constraint
between the Neumann-like system and the negative order MKdV hierarchy, all
equations in the hierarchy are proven to have the parametric representations of
solutions. In particular, we obtain the parametric solutions of the sine-Gordon
equation and the sinh-Gordon equation.
