How to parameterize algebraic dependencies?

We propose the unified and geometric formulation to the uniformization theory of Riemann surfaces and orbifolds of finite genera. Complete description is based on the standard Fuchsian differential equations but is extended to the Abelian integrals and analytic connection on a cotangent bundle over the surfaces/orbifolds. The invariant (geometric) description reduces to a fundamental system of differential equations for a set of holomorphic integrals. We exhibit the first explicitly solvable example: the case when holomorphic integrals on a Riemann surface of genus $g=2$, as functions of the uniformizing variable, is analytically representable in terms of known functions.