  A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun.
The existence problem for $k$-sun decompositions of $K_v$, with $k$ odd, has
been solved only when $k=3$ or $5$.
  By adapting a method used by Hoffmann, Lindner and Rodger to reduce the
spectrum problem for odd cycle systems of the complete graph, we show that if
there is a $k$-sun system of $K_v$ ($k$ odd) whenever $v$ lies in the range
$2k< v < 6k$ and satisfies the obvious necessary conditions, then such a system
exists for every admissible $v\geq 6k$.
