Finite-gap Solutions of the Fuchsian Equations

We find a new class of the Fuchsian equations, which have an algebraic geometric solutions with the parameter belonging to a hyperelliptic curve. Methods of calculating the algebraic genus of the curve, and its branching points, are suggested. Numerous examples are given. Introduction Integrability of the Heun equation with half-odd characteristic exponents dy dz + P (z) dy dz +Q(z)y = 0, (0.1) where P (z) = 1 2 ( 1− 2m1 z + 1− 2m2 z − 1 + 1− 2m3 z − a ) , (0.2) Q(z) = N(N − 2m0 − 1)z + λ 4z(z − 1)(z − a) , (0.3) N = m0 +m1 +m2 +m3, mi ∈ Z>0, λ, z ∈ C, (0.4) was probably discovered by Darboux more then 100 years ago [1]. But only recently so-called finite-gap solutions Y1,2(m;λ; z) = √ Ψg,N(λ, z) exp ( ± iν(λ) 2 ∫ z1(z − 1)2(z − a)3 dz Ψg,N(λ, z) √ z(z − 1)(z − a) ) (0.5) of this Heun equation were wrote out and analyzed [2, 3]. Here i = −1,