  We present two examples of reductions from the evolution equations describing
discrete Schlesinger transformations of Fuchsian systems to difference
Painlev\'e equations: difference Painlev\'e equation
d-$P\left({A}_{2}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{6}$ and
difference Painlev\'e equation d-$P\left({A}_{1}^{(1)*}\right)$ with the
symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to
compute their Okamoto space of the initial conditions and emphasize the role
played by geometry in helping us to understand the structure of the reduction,
a choice of a good coordinate system describing the equation, and how to
compare it with other instances of equations of the same type.
