Vector systems of Painlev\'e type

Abstract

The group reduction procedure is applied to vector generalizations of the NLS, mKdV, and KdV equations. The resulting ODE systems admit isomonodromic Lax representations and are multicomponent generalizations of the Painlev\'e equations P$_1$, P$_2$, P$_{34}$, and P$_4$. Some of them can be interpreted as nonautonomous deformations of well-known systems integrable in the Liouville sense, in particular, the Garnier and H\'enon--Heiles systems. In one case, an unexpected connection with the equations of quasiperiodic dressing chain for the Schr\"odinger operator is established.