Finite Integrable System And Painlevé Analysis

The finite integrable system is important in history of differential equation. Thediscovery of finite integrable system usually depends on some special skills, thereforethe finite integrable systems are very few. There exists lot of soliton equations asinfinite integrable system in other hand. The nonlinearization method establishes arelation between the finite integrable system and the infinite integrable system. In thispaper we consider a two order matrix spectral problem. A finite integrable system isdiscovered by a nonlinear constrain. Ninteen solutions of a soliton equation areobtained by means of Painlevé analysis. In the end of the paper, a finite integrablesystem is reduced from stationary MKdV equation. The MKdV equation is alsowritten as a finite integrable system in the Jacobi-Ostrogradski’s coordinate.