Critical metric for an Hilbert action?

Suppose that $$\omega$$ is an 1-form on a Reimannain Manifold $$(M,g)$$ and $$s$$ is a $$(0,2)$$ symmetric tensor which be considered as $$(1,1)$$ symmetric tensor whenever it is convenient. If for all $$s$$ the following integral vanishes,

$$\int_M <‎\big( ‎‎-\mathrm{Ric} ‎+‎‎\dfrac{1}{2}‎Rg+‎‎2\mathrm{Sym}‎‎(\nabla ‎\omega)‎ +\omega\otimes \omega -|\pi |^2 ‎g\big) ‎,s>dV_‎g\\ +2\int_M (+<‎\vec{\nabla}\mathrm{tr}(s),‎U> )‎dV_‎g =0$$

Then what we can obtain from this equality?

Note that $$U$$ is is defined by $$w=g(U,.)$$.