  F(R) theory of gravity is claimed to admit a host of conserved currents under
the imposition of Noether symmetry following various techniques. However, for a
constrained system such as gravity, Noether symmetry is not on-shell. As a
result, the symmetries do not necessarily satisfy the field equations in
general, constraints in particular, unless the generator is modified to
incorporate the constraints. In the present manuscript, we apply the first
theorem of Poisson to unveil the fact that not all the conserved currents
appearing in the literature for F(R) theory of gravity, satisfy the field
equations. We also provide a list of available forms of F(R) along with
associated conserved currents, and construct a generalized action, which might
address the cosmic puzzle, elegantly.
