  We show how the Abel-Jacobi map provides all the principal properties of an
ample family of integrable mechanical systems associated to hyperelliptic
curves. We prove that derivative of the Abel-Jacobi map is just the St\"{a}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 to construct several flat coordinate systems on a common hyperelliptic
curve and to connect among themselves different integrable systems on a single
phase space.
