  We develop a procedure to reproduce the ten-dimensional generalized
supergravity equations from T-duality covariant equations, that facilitates
generalization to U-duality covariant formulations of eleven-dimensional
supergravity. The latter leads to a modification of the eleven-dimensional
supergravity equations with terms that contain a rank-2 tensor field $J^{mn}$
which is the eleven-dimensional analog of the non-unimodularity Killing vector
$I^m$ in ten dimensions.
