  An $\ell$-cycle decomposition of $K_v$ is said to be \emph{equitably
$2$-colourable} if there is a $2$-vertex-colouring of $K_v$ such that each
colour is represented (approximately) an equal number of times on each cycle:
more precisely, we ask that in each cycle $C$ of the decomposition, each colour
appears on $\lfloor \ell/2 \rfloor$ or $\lceil \ell/2 \rceil$ of the vertices
of $C$. In this paper we study the existence of equitably 2-colourable
$\ell$-cycle decompositions of $K_v$, where $\ell$ is odd, and prove the
existence of such a decomposition for $v \equiv 1, \ell$ (mod $2\ell$).
