The Killing condition $\nabla_a X^\flat_b + \nabla_b X^\flat_a = 0$, together with the gradient condition (really, the closedness condition for a $1$-form) $\nabla_a X^\flat_b - \nabla_b X^\flat_a = 0$, imply that $X$ is covariantly constant, $\nabla_a X^b = 0$. Note that I'm using the relation $X^\flat(-) = \langle X, - \rangle$ to define the $1$-form $X^\flat$.

The existence of covariantly constant vectors implies various restrictive conditions on the geometry of the Riemannian manifold $(M^n, g)$. If there's any particular restriction that you are interested in, you may want to refine your question.