How can I prove that the language $L=\left\{\langle M\rangle\mid L(M)=\left\{\langle M\rangle\right\}\right\}$ is not decidable?

When trying to use a diagonal argument, I cannot conclude from $L(M)\ne\left\{\langle M\rangle\right\}$ that $\langle M\rangle\not\in L(M)$.

Also, it seems that I cannot apply Rice's theorem because I can't say anything about the machines' encodings $\langle M\rangle$ in the definition of the set $\mathcal{S}$.

Is there an easy reduction to a known undecidable language?