How do the spin-matrices change in systems without spherical symmetry?

The (non-relativistic) spin matrices of a free electron, with $z$ as the quantization axis, read:
$$
\boldsymbol{S}\_{x}=\frac{\hbar}{2}\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right], \quad \boldsymbol{S}\_{y}=\frac{\hbar}{2}\left[\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right], \quad \boldsymbol{S}\_{z}=\frac{\hbar}{2}\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right].
$$
The eigenvalues of these matrices are related to the energies when measuring the particle in a particular spin state. The spherical symmetry is reflected in $\boldsymbol{S}\_x$ and $\boldsymbol{S}\_y$ which have the same energy eigenvalues. My question now is, what happens if we consider a doublet molecule *without* symmetry along the $x$- and $y$-directions? Will $\boldsymbol{S}\_x$ and $\boldsymbol{S}\_y$ then no longer have the same eigenvalues?


