First, we describe a classification of the real valued solutions of the Toda equation in terms of their parabolic weights, from the viewpoint of the Kobayashi-Hitchin correspondence for wild harmonic bundles. Then, we discuss when the associated semi-infinite variation of Hodge structure has an integral structure.
It follows from two results. One is the explicit computation of the Stokes factors of a certain meromorphic flat bundle. The other is an explicit description of the associated meromorphic flat bundle.
We use the opposite filtration of the limit mixed twistor structure with an induced torus action.