  We determine the generic complete eigenstructures for $n \times n$ complex
symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More
precisely, we show that the set of $n \times n$ complex symmetric matrix
polynomials of odd grade $d$, i.e., of degree at most $d$, and rank at most $r$
is the union of the closures of the $\lfloor rd/2\rfloor+1$ sets of symmetric
matrix polynomials having certain, explicitly described, complete
eigenstructures. Then, we prove that these sets are open in the set of $n
\times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at
most $r$. In order to prove the previous results, we need to derive necessary
and sufficient conditions for the existence of symmetric matrix polynomials
with prescribed grade, rank, and complete eigenstructure, in the case where all
their elementary divisors are different from each other and of degree $1$. An
important remark on the results of this paper is that the generic
eigenstructures identified in this work are completely different from the ones
identified in previous works for unstructured and skew-symmetric matrix
polynomials with bounded rank and fixed grade larger than one, because the
symmetric ones include eigenvalues while the others not. This difference
requires to use new techniques.
