  We show that the set of $n \times n$ complex symmetric matrix pencils of rank
at most $r$ is the union of the closures of $\lfloor r/2\rfloor +1$ sets of
matrix pencils with some, explicitly described, complete eigenstructures. As a
consequence, these are the generic complete eigenstructures of $n \times n$
complex symmetric matrix pencils of rank at most $r$. We also show that these
closures correspond to the irreducible components of the set of $n\times n$
symmetric matrix pencils with rank at most $r$ when considered as an algebraic
set.
