  We study the effective D=4, N=1 supergravity description of five-dimensional
heterotic M-theory in the presence of an M5 brane, and derive the Killing
vectors and isometry group for the Kahler moduli-space metric. The group is
found to be a non-semisimple maximal parabolic subgroup of Sp(4,R), containing
a non-trivial SL(2,R) factor. The underlying moduli-space is then naturally
realised as the group space Sp(4,R)/U(2), but equipped with a nonhomogeneous
metric that is invariant only under that maximal parabolic group. This
nonhomogeneous metric space can also be derived via field truncations and
identifications performed on Sp(8,R)/U(4) with its standard homogeneous metric.
In a companion paper we use these symmetries to derive new cosmological
solutions from known ones.
