  The relationship of two dimensional quantum field theory and isomonodromic
deformations of Fuchsian systems has a long history. Recently four-dimensional
$\mathcal{N}=2$ gauge theories joined the party in a multitude of roles. In
this paper we study the vacuum expectation values of intersecting half-BPS
surface defects in $SU(2)$ theory with $N_f=4$ fundamental hypermultiplets. We
show they form a horizontal section of a Fuchsian system on a sphere with $5$
regular singularities, calculate the monodromy, and define the associated
isomonodromic tau-function. Using the blowup formula in the presence of
half-BPS surface defects, initiated in the companion paper, we obtain the GIL
formula, establishing an unexpected relation of the topological string/free
fermion regime of supersymmetric gauge theory to classical integrability.
