  We consider dimensional reduction of the eleven-dimensional supergravity to
less than four dimensions. The three-dimensional $E_{8(+8)}/SO(16)$ nonlinear
sigma model is derived by direct dimensional reduction from eleven dimensions.
In two dimensions we explicitly check that the Matzner-Misner-type $SL(2,R)$
symmetry, together with the $E_8$, satisfies the generating relations of $E_9$
under the generalized Geroch compatibility (hypersurface-orthogonality)
condition. We further show that an extra $SL(2,R)$ symmetry, which is newly
present upon reduction to one dimension, extends the symmetry algebra to a real
form of $E_{10}$. The new $SL(2,R)$ acts on certain plane wave solutions
propagating at the speed of light. To show that this $SL(2,R)$ cannot be
expressed in terms of the old $E_9$ but truly enlarges the symmetry, we
compactify the final two dimensions on a two-torus and confirm that it changes
the conformal structure of this two-torus.
