A reduction of the spectrum problem for odd sun systems and the prime case

A k-cycle with a pendant edge attached to each vertex is called a k-sun. The existence problem for k-sun decompositions of Kv, 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 (Formula presented.) -sun system of (Formula presented.) ((Formula presented.) odd) whenever (Formula presented.) lies in the range (Formula presented.) and satisfies the obvious necessary conditions, then such a system exists for every admissible (Formula presented.). Furthermore, we give a complete solution whenever k is an odd prime.