TITLE: When do isometric actions exist?
QUESTION [10 upvotes]: Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the action of $G$ becomes an isometric action? Conversely, given a metric on $X$, is there a nontrivial action of $G$ on $X$ that preserves the metric?
I am looking for the most general necessary and sufficient conditions and any possible obstructions. For the first question, the answer is obviously positive when $G$ is compact: one chooses a metric $d$ on $X$ and simply does an "averaging process along the orbits" by defining
$$ \rho(x,y) = \int_{G} d(g^{-1}\cdot x,g^{-1}\cdot y) dg . $$
I suspect that a similar idea would work more generally using a "cut-off" function on $X$ when the action of $G$ on $X$ is proper. Any connections to amenability (of the group or the group action) would also be interesting.

REPLY [9 votes]: I am sorry That I am getting involved so late. I was away at a meeting for two days. I have an old (1961) paper in the ANNALS OF MATH called "On the Existence of Slices for Actions of Non-Compact Lie Groups" which is quite relevant. In particular, on page 318 you can find the following concerning proper actions of an arbitrary Lie group $G$. 
Theorem 4.3.4. Every seperable, metrizeable, proper $G$-space $X$ admits an invariant metric. ...
There are a large number of other theorems there that show that the theory of proper G-spaces for G an arbitrary Lie group is similar to the theory of G-spaces for a compact Lie group. You can find the paper here: http://vmm.math.uci.edu/ExistenceOfSlices.pdf