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We study Rainich-like conditions for symmetric and trace-free tensors T. Forarbitrary even rank we find a necessary and sufficient differential conditionfor a tensor to satisfy the source free field equation. For rank 4, in ageneric case, we combine these conditions with previously obtained algebraicconditions to obtain a complete set of algebraic and differential conditions onT for it to be a superenergy tensor of a Weyl candidate tensor satisfying theBianchi vacuum equations. By a result of Bell and Szekeres this implies that invacuum, generically, T must be the Bel-Robinson tensor of the spacetime. Forthe rank 3 case we derive a complete set of necessary algebraic anddifferential conditions for T to be the superenergy tensor of a massless spin3/2 field satisfying the source free field equation.Comment: 14 pages, with corrections, to appear in Proc. Roy. Soc. London

Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank