  We study 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 obtain 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.
