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Integrability of second-order Fuchsian equation on the torus \(T^ 2\). (English) Zbl 0856.34008
Consider the differential equation \((*)\) \(u''- \lambda\wp(t) u= 0\), where \(\lambda\) denotes a positive real parameter, \(u(t)\) is a complex-valued function of a complex variable \(t\) and \(\wp(t)\) is the elliptic function of Weierstrass with periods \(w_1= 2\alpha\) and \(w_2= 2\alpha i\) (\(\alpha\) real). For \(\lambda= n(n- 1)\) \((n= 1, 2,\dots)\), the authors establish the following: Firstly, each solution of equation \((*)\) is a meromorphic function on \(\mathbb{C}\). Next, by regarding \((*)\) as a Fuchsian equation on a torus \(T^2\) (constructed from the period parallelogram of \(\wp(t)\)) which possesses a unique regular singularity on \(T^2\), they define the monodromy group of \((*)\) showing that it is solvable, and then extend the concept of integrability for \((*)\) from the Riemann sphere onto the torus \(T^2\).
Another result for the special case \(\lambda= 6\) is also proved.
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations