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A lecture on the octahedron. (English) Zbl 1044.34050

Algebraic solutions to the classical Painlevé sixth equation PVI are explicitly constructed by solving linear systems with monodromy contained in the octahedral subgroup of $SO\left(3\right)$.

The author starts transforming the conventional linear $2×2$-matrix ODE associated to PVI into a 3-vector form. Thus the relevant monodromy group becomes a subgroup of $SO\left(3,ℂ\right)$. The author restricts himself to the case where the local monodromy matrices are all rotations with the same angle, and considers non-abelian subgroups of the octahedral group promising to treat the icosahedral case elsewhere. In each of the octahedral, tetrahedral and dihedral cases, the author constructs an abstract Riemann surface and its quotients which are mapped by a solution to the above linear $3×3$ system to algebraic curves. The latter associate to certain classical problems in projective geometry. Solving them, the author extracts the parametric description of the algebraic solutions to PVI.

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 14H45 Special curves and curves of low genus