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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.
##### MSC:
 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