= Solution
Write $|\alpha\rangle=a|0\rangle+b|1\rangle$. Expanding in the orthonormal <Bell state>[Bell basis] of systems $CA$ gives the <Bell-basis teleportation identity>
$$
|\alpha\rangle_C|\beta_{00}\rangle_{AB}
=\frac12\bigl(
|\beta_{00}\rangle_{CA}|\alpha\rangle_B
+|\beta_{01}\rangle_{CA}Z|\alpha\rangle_B
+|\beta_{10}\rangle_{CA}X|\alpha\rangle_B
+|\beta_{11}\rangle_{CA}XZ|\alpha\rangle_B
\bigr).
$$
Thus, uniformly for $x,z\in\{0,1\}$,
$$
\boxed{|\mu_{xz}\rangle=X^xZ^z|\alpha\rangle}.
$$
The states are unique because the four Bell states are an <orthonormal basis> of the two-qubit space. In particular,
$$
\boxed{|\mu_{10}\rangle=X|\alpha\rangle}.
$$
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