We study Stokes phenomena of the k \times k isomonodromy systems with an
arbitrary Poincar\'e index r, especially which correspond to the
fractional-superstring (or parafermionic-string) multi-critical points (\hat
p,\hat q)=(1,r-1) in the k-cut two-matrix models. Investigation of this system
is important for the purpose of figuring out the non-critical version of M
theory which was proposed to be the strong-coupling dual of fractional
superstring theory as a two-matrix model with an infinite number of cuts.
Surprisingly the multi-cut boundary-condition recursion equations have a
universal form among the various multi-cut critical points, and this enables us
to show explicit solutions of Stokes multipliers in quite wide classes of
(k,r). Although these critical points almost break the intrinsic Z_k symmetry
of the multi-cut two-matrix models, this feature makes manifest a connection
between the multi-cut boundary-condition recursion equations and the structures
of quantum integrable systems. In particular, it is uncovered that the Stokes
multipliers satisfy multiple Hirota equations (i.e. multiple T-systems).
Therefore our result provides a large extension of the ODE/IM correspondence to
the general isomonodromy ODE systems endowed with the multi-cut boundary
conditions. We also comment about a possibility that N=2 QFT of Cecotti-Vafa
would be "topological series" in non-critical M theory equipped with a single
quantum integrability.Comment: 43 pages, 3 figures; v2:references and comments added (footnote 24
