The isomonodromic deformation plays a universal role in
connecting various research areas of mathematics and physics. In this
talk, I am going to discuss the isomonodromy theory for a new class of
Fuchsian-type elliptic second-order equations defined on the moduli
space of elliptic curves with the parameter τ ∈ H (upper half-plane).
We will observe that the isomonodromic deformation equation is
governed by a new second-order nonlinear equation with a deep
connection to the Painlevé VI equation but admits essentially
different properties from PVI. Indeed, the new isomonodromic
deformation equation, distinct from the Painlevé VI equation, admits
so-called movable branch points that can be explicitly determined by
their monodromy data.