Negative order MKdV hierarchy and a new integrable Neumann-like system
  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. 
Zhijun Qiao