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We list all metrics of arbitrary signature in four dimensions which admitcomplete separation of variables in the Hamilton--Jacobi equation for geodesicHamiltonians. There are only ten classes of separable metrics admittingcommuting Killing vector fields, indecomposable quadratic conservation laws,and coisotropic coordinates. Canonical separable metrics parameterized byseveral (up to twelve) arbitrary functions of single coordinates are writtenexplicitly. The full set of independent conservation laws in involution foreach canonical metrics is also found.

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