We present a study of Rainich-like conditions for symmetric and trace-free tensors
T
. For arbitrary even rank we find a necessary and sufficient differential condition for a tensor to satisfy the source-free field equation. For rank 4, in a generic case, we combine these conditions with previously obtained algebraic conditions to gain a complete set of algebraic and differential conditions on
T
for it to be a superenergy tensor of a Weyl candidate tensor, satisfying the Bianchi vacuum equations. By a result of Bell and Szekeres, this implies that in vacuum, generically,
T
must be the Bel–Robinson tensor of the spacetime. For the rank 3 case, we derive a complete set of necessary algebraic and differential conditions for
T
to be the superenergy tensor of a massless spin-3/2 field, satisfying the source-free field equation.
A differential condition on the normal cone
