Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

Let $\pi$ and $\pi'$ be two distinct partitions of an odd integer $v$ into parts $\geq 3$, and let $F$ and $F'$ be $2$-regular spanning subgraphs of the complete graph $K_{v}$ whose lists of cycle-lengths are $\pi$ and $\pi'$, respectively. The Hamilton-Waterloo problem HWP($v; \pi, \pi'; r, r'$) asks for a decomposition of $K_{v}$ into $r$ copies of $F$ and $r'$ copies of $F'$, with $r+r'=(v-1)/2$. As far as we are aware, very little is known on the existence of solutions to HWP when both $\pi$ and $\pi'$ contain only odd terms, and not many of the known solutions to HPW exhibit algebraic regularity. In this paper, we prove that HWP$(\ell(2\ell n+1);[\ell^{2\ell n+1}],[(2\ell n+1)^\ell];\ell n,$ ${\frac{(\ell-1)(2\ell n+1)}{2}})$ has a solution whenever $\ell \equiv 1\pmod{4}$ and $n\geq (\ell-1)/2$. Moreover, these solutions possess a cyclic automorphism group with a sharply-transitive action on the vertex set.