  The paper discusses P$_{III-V}$ equation for special values of its parameters
for which this equation reduces to P$_{III}$, I$_{12}$, as well as, to some
special cases of I$_{38}$ and I$_{49}$ equations from the Ince's list of $50$
second order differential equations possessing Painlev\'e property.
  These reductions also yield symmetries governing the reduced models obtained
from the P$_{III-V}$ equation. We point out that the solvable equations on
Ince's list emerge in this reduction scheme when the underlying reflections of
the Weyl symmetry group no longer include an affine reflection through the
hyperplane orthogonal to the highest root and therefore do not give rise to an
affine Weyl group. We hypothesize that on the level of the underlying algebra
and geometry this might be a fundamental feature that distinguishes the six
Painlev\'e equations from the remaining $44$ solvable equations on the Ince's
list.
