  Linear zero-rest-mass fields generically develop logarithmic singularities at
the critical sets where spatial infinity meets null infinity. Friedrich's
representation of spatial infinity is ideally suited to study this phenomenon.
These logarithmic singularities are an obstruction to the smoothness of the
zero-rest-mass field at null infinity and, in particular, to peeling. In the
case of the spin-2 field it has been shown that these logarithmic singularities
can be precluded if the initial data for the field satisfies a certain
regularity condition involving the vanishing, at spatial infinity, of a certain
spinor (the linearised Cotton spinor) and its totally symmetrised derivatives.
In this article we investigate the relation between this regularity condition
and the staticity of the spin-2 field. It is shown that while any static spin-2
field satisfies the regularity condition, not every solution satisfying the
regularity condition is static. This result is in contrast with what happens in
the case of General Relativity where staticity in a neighbourhood of spatial
infinity and the smoothness of the field at future and past null infinities are
much more closely related.
