On the second order analytic system with four singular points

In the work second order systems of differential equations with meromorphic coefficients are considered. It is known that when singular points of system are first order poles, such system of equations has canonical form. Main object of investigation is regular system of equations on Riemann sphere, when number of singular points are four, class of such systems contains well known Heun equation. The structure of work is following. In the introduction and first section given necessary definitions and auxiliary constructions. In particular, the geometric and analytic characterization of solution spaces of hypergeometric and Riemann equations geometrically and analytically point of view is considered. Second section dedicated investigation of regular system with four singular pints and gives some results, including Schwarz-Cristoffel theorem, next this results used for proof of main result of the work when given final second section. Proved that conformal moduli of ordered singular points is invariant for Heun equation. More exactly, proved that if two ordered selection of four points are conformal equivalent and corresponding systems of equations induced from Heun equations with different singular points are gauge equivalent, then Riemann data coincides. This result permits generalization for more general systems of regular equations.

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