Everywhere invariant spaces of metrics and isometries

We define an everywhere invariant space of metricsU to be one for which the set of diffeomorphisms, which leaveU invariant, contains all the isometries of the individual metrics inU. We also generalize Killing's equation to a new equation, the invariance equation, which has as solutions those vector fields which leaveU invariant. By combining these ideas we give a new method for finding the isometries of a given metric.