Show $L_U \le_m S$
Let $v=\langle T, w\rangle$, where $T$ is a Turing machine and $w$ is an input string to $T$. Fix two different input symbols $a$ and $b$. Construct Turing machine $M_v$ that upon input $x$, check whether $x$ is $aww^Rb$.
- If it is, accept.
- Otherwise, check whether $x$ is $bww^Ra$.
- If it is, simulate $T$ upon input $w$.
- Otherwise, accept.
It is straightforward to check that $v\in L_U$ $\iff$ $\langle M_v\rangle\in S$. The construction from $v$ to $M_v$ is computable. Hence $L_U \le_m S$
Show $L_U \le_m \bar S$
Let $v=\langle T, w\rangle$, where $T$ is a Turing machine and $w$ is an input string to $T$. Fix two different input symbols $a$ and $b$. Construct Turing machine $M_v$ that upon input $x$, check whether $x$ is $aww^Rb$.
- If it is, loop forever.
- Otherwise, check whether $x$ is $bww^Ra$.
- If it is, simulate $T$ upon input $w$.
- Otherwise, accept.
It is straightforward to check that $v\in L_U$ $\iff$ $\langle M_v\rangle\in \bar S$. The construction from $v$ to $M_v$ is computable. Hence $L_U \le_m \bar S$