On the spouse-loving variant of the Oberwolfach problem

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 $m\ge 5$ 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 $m_1,\dots ,m_t$, whenever $m_1,\dots ,m_t$ are all even.