  Motivated by the recent analysis of the E10 sigma model for the study of M
theory, we study a one-dimensional sigma model associated with the hyperbolic
Kac-Moody algebra G2H and its link to D=5, N=2 pure supergravity, which closely
resembles in many ways D=11 supergravity. The bosonic equations of motion and
the Bianchi identity for D=5 pure supergravity match the equations of the level
l<=3 truncation of the G2H sigma model up to higher level terms, just as they
do for the D=11 case. We also compute low level root and outer multiplicities
in the A3 decomposition, and indeed find singlets at l=4k, k=2,3,...
corresponding to the scaling of ER^{k+1} terms, although the missing singlet at
l =4 remains a puzzle.
