  We examine the structure of the cotangent bundle $T^{*}X$ of an algebraic
variety $X$ acted on by a reductive group $G$ from the viewpoint of equivariant
symplectic geometry. In particular, we construct an equivariant symplectic
covering of $T^{*}X$ by the cotangent bundle of a certain variety of
horospheres in $X$, and integrate the invariant collective motion on $T^{*}X$.
These results are based on a "local structure theorem" describing the action of
a certain parabolic in $G$ on an open subset of $X$, which is interesting by
itself.
