  Generalizing previous work by two of us, we prove the non-existence of
certain stationary configurations in General Relativity having a spatial
reflection symmetry across a non-compact surface disjoint from the matter
region. Our results cover cases such that of two symmetrically arranged
rotating bodies with anti-aligned spins in $n+1$ ($n \geq 3$) dimensions, or
two symmetrically arranged static bodies with opposite charges in 3+1
dimensions. They also cover certain symmetric configurations in
(3+1)-dimensional gravity coupled to a collection of scalars and abelian vector
fields, such as arise in supergravity and Kaluza-Klein models. We also treat
the bosonic sector of simple supergravity in 4+1 dimensions.
