Complete separation of variables in the geodesic Hamilton--Jacobi equation in four dimensions

We list all metrics of arbitrary signature in four dimensions which admit complete separation of variables in the Hamilton–Jacobi equation for geodesic Hamiltonians. There are only ten classes of separable metrics admitting commuting Killing vector fields, indecomposable quadratic conservation laws, and coisotropic coordinates. Canonical separable metrics parameterized by several (up to twelve) arbitrary functions of single coordinates are written explicitly. The full set of independent conservation laws in involution for each canonical metrics is also found.