  We prove that $K_n+I$, the complete graph of an even order with a $1$-factor
duplicated, admits a decomposition into $2$-factors, each a disjoint union of
cycles of length $m \geq 5$ if and only if $m \mid n$, except possibly when $m$
is odd and $n=4m$. In addition, we show that $K_n+I$ admits a decomposition
into $2$-factors, each a disjoint union of cycles of lengths $m_1, \ldots,
m_t$, whenever $m_1, \ldots, m_t$ are all even.
