out the intrinsic relationship.

In FQHEs, the Laughlin trial wavefunction reveals several remarkable properties of FQHEs at filling number $\nu=\frac{1}{2m+1},$ such as the fractional statistics as well as the topological orders. Later on a conformal field theory (CFT) realization was discovered which shows that the wavefunction is corresponding to a correlation function of certain vertex operators . Furthermore this idea is generalized to many other FQH states, e.g. Halperin state[@halperin1984statistics], Moore-Read state, and Read-Rezayi state[@read1999beyond], et.al[@bergholtz2008quantum; @gurarie1997haldane; @moore1991nonabelions; @wen1994chiral]. However, CFT is possibly not sufficient to drive the dynamics of the edge theory, since it only determines the behavior of the theory near critical point.[^1].

A more ambitious thinking is to find the Hamiltonian system behind the edge ground state. So far, there are two classes of Hamiltonian analysis for FQHE. One is the Chern-Simons approach, initiated by Zhang, Hansson and Kivelson in 1989[@zhang1989effective]. The other is the extended Hamiltonian theory, introduced by Murthy and Shankar in late 90’s[@murthy1999hamiltonian; @murthy2003hamiltonian; @shankar1997towards]. The later one contains Chern-Simons as its asymptotic theory.

In our study we try to approach the integrability problem in a different way. In fact, we are not meant to establish a unified Hamiltonian theory to solve the complicated 