  We explore some new off-shell and on-shell conserved quantities for a scalar
field in Minkowski space, using integrability condition. The off-shell
conserved tensors are related to the kinematics of the field, while a linear
combination of the off-shell and the on-shell conserved tensors ends up with
the energy-momentum tensor for the scalar field. In the curved background,
using Ricci and Bianchi identities, Brans-Dicke type field equations emerge,
without requiring the principle of equivalence. Further, starting from the
curvature scalar and using these identities, the field equations for modified
gravity (Einstein-Hilbert action in the presence of higher-order terms)
follows.
