  Given a locally compact Polish space X, a necessary and sufficient condition
for a group G of homeomorphisms of X to be the full isometry group of (X,d) for
some proper metric d on X is given. It is shown that every locally compact
Polish group G acts freely on GxY as the full isometry group of GxY with
respect to a certain proper metric on GxY, where Y is an arbitrary locally
compact Polish space with (card(G),card(Y)) different from (1,2). Locally
compact Polish groups which act effectively and almost transitively on complete
metric spaces as full isometry groups are characterized. Locally compact Polish
non-Abelian groups on which every left invariant metric is automatically right
invariant are characterized and fully classified. It is demonstrated that for
every locally compact Polish space X having more than two points the set of
proper metrics d such that Iso(X,d) = {id} is dense in the space of all proper
metrics on X.
