Main objects of uniformization of the curve y 2 = x 5 − x are studied: its Burnside's parametrization, corresponding Schwarz's equation, and accessory parameters. As a result we obtain the first examples of solvable Fuchsian equations on torus and exhibit number-theoretic integer q-series for uniformizing functions, relevant modular forms, and analytic series for holomorphic Abelian integrals. A conjecture of Whittaker for hyperelliptic curves and its hypergeometric reducibility are discussed. We also consider the conversion between Burnside's and Whittaker's uniformizations.

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We represent and analyze the general solution of the sixth Painlevé transcendent P6 in the Picard-HitchinOkamoto class in the Painlevé form as the logarithmic derivative of the ratio of τ -functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the P6 equation and the uniformization of algebraic curves and present examples.

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Until recently, the following question has not been considered: how to generalise constructions, related to the Dubrovin's equations (DE) in finite-gap potential theory for the Schrödinger operator, into arbitrary spectral problems. We have in mind the equations of the zeros of the Ψ-function and the trace formulas. In spite of physical interpretation of these objects as analogs of the scattering data, the generalizations are not clear, or may even be absent. This problem has an independent interest. For example, the well known Novikov's equations, appearing in a general theory of finite-gap integration, are (to all appearences) completely integrable finite-dimensional dynamical systems. A reduction of the DE to the Jacobi inverse problem demonstates the Liouville's integrability of these equations. Note that if an algebraic curve is a cover over an elliptic curve, then one can use the trace formulas to produce solutions in elliptic functions, new finite-gap potentials and some applications to nonlinear integrable partial differential equations.In a recent note [2] an universal feature of the finite-gap potentials was revealed: they form a class, which admits an integration of the spectral problem by quadratures. It was shown there how to obtain all the ingredients of the straight spectral problem: the Ψ-formula, algebraic curve, Novikov's equations and their integrals. On the other hand, as soon as Ψ is known, it is natural to expect that the equations for its zeroes γ k (x) may be written using simple arguments. This can be done, and we show algorithmically how to solve the problem with the appearance of related objects: trace formulas and the Abel transformation. We do not discuss here a separate question about an exact (or one-to-one) correspondence between the following two constructions:• an algebraic curve and the divisor of the zeroes {γ k };• boundary conditions of the Dirichlet type Ψ(x o ) = Ψ(x o + Ω) = 0;