Mathematics

We show that the fourth order nonlinear ODE which controls the pole dynamics
in the general solution of equation $P_I^2$ compatible with the KdV equation
exhibits two remarkable properties: 1) it governs the isomonodromy deformations
of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2) it does
not possesses the Painlev\'e property. We also study the properties of the
Riemann--Hilbert problem associated to this ODE and find its large $t$
asymptotic solution for the physically interesting initial data.