Let $X$ be set. Let $f$ be a function from $X$ into $X$. For a given set $E\subseteq X$, we say $E$ determines a $U$-part of $f$ if $f(E)\subseteq E$ and the restriction $f:E\to E$ is a bijection.

I am looking for a commuting pair of ** injective** functions $f:X\to X$ and $g:X\to X$ (I mean $fg=gf$) such that both $f$ and $g$ have $U$-parts but not common $U$-part.