  We continue the study of the one-dimensional E10 coset model (massless
spinning particle motion on E10/K(E10) whose dynamics at low levels is known to
coincide with the equations of motion of maximal supergravity theories in
appropriate truncations. We show that the coset dynamics (truncated at levels
less or equal to three) can be consistently restricted by requiring the
vanishing of a set of constraints which are in one-to-one correspondence with
the canonical constraints of supergravity. Hence, the resulting constrained
sigma-model dynamics captures the full (constrained) supergravity dynamics in
this truncation. Remarkably, the bosonic constraints are found to be
expressible in a Sugawara-like (current x current) form in terms of the
conserved E10 Noether current, and transform covariantly under an upper
parabolic subgroup E10+ of E10. We discuss the possible implications of this
result, and in particular exhibit a tantalising link with the usual affine
Sugawara construction in the truncation of E10 to its affine subgroup E9.
