  We introduce equivariant versions of uniform rationality: given an algebraic
group G, a G-variety is called G-uniformly rational (resp. G-linearly uniformly
rational) if every point has a G-invariant open neighborhood equivariantly
isomorphic to a G-invariant open subset of the affine space endowed with a
G-action (resp. linear G-action). We establish a criterion for Gm-uniform
rationality of affine variety equipped with hyperbolic Gm-action with a unique
fixed point, formulated in term of their Altmann-Hausen presentation. We prove
the Gm-uniform rationality of Koras-Russel threefolds of the first kind and we
also give example of non Gm-uniformly rational but smooth rational Gm-threefold
associated to pairs of plane rational curves birationally non equivalent to a
union of lines.
