Let (M,g) be a Riemannian manifold with the metric g of any signature. The authors consider the Hamiltonian H on \(T^*M\) of the type \(H=g^{ij}p_ ip_ j+V\quad\) \((g^{ij}\) are contravariant components of g, V is a function on M) and study the corresponding Hamilton-Jacobi equation. For H satisfying Levi-Cività separability conditions, suitable coordinates in charts (called normal separable coordinates) are constructed, in which \(g^{ij}\) and V assume an ”irreducible” form. This leads to complete integrability of the Hamilton-Jacobi equation, which is obtained by the integration of the correspondent separated system of ordinary differential equations in normal separable coordinates.