We show how the Abel-Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hypeielliptic curves. We prove that derivative of the Abel-Jacobi map is just the St?ckel matrix, which determines n-orthogonal curvilinear coordinate systems in a flat space. The Lax pairs, r-matrix algebras and explicit form of the flat coordinates are constructed. An application of the Weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space. ? 1999 American Institute of Physics.