On special limits of the Mixed Painlevé PIII-V Model

The paper discusses PIII-V equation for special values of its parameters for which this equation reduces to PIII, I12, as well as, to some special cases of I38 and I49 equations from the Ince's list of 50 second order differential equations possessing Painlevé property. These reductions also yield symmetries governing the reduced models obtained from the PIII-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 Painleve equations from the remaining 44 solvable equations on the Ince's list.