New conserved tensors and Brans-Dicke type field equation, using integrability condition

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.