On linearised vacuum constraint equations on Einstein manifolds

We show how to parameterise solutions of the general relativistic vector constraint equation on Einstein manifolds by unconstrained potentials. We provide a similar construction for the trace-free part of tensors satisfying the linearised scalar constraint. Previous work of ours has provided similar different constructions for solutions of the linearized constraints in the case where the cosmological constant $\mathrm{\Lambda }$ is zero. We use our new potentials to show that one can shield linearised gravitational fields using linearised gravitational fields without imposing the TT gauge (as done in previous work), for any value of $\mathrm{\Lambda }\in \mathbb{R}$.