  We study the $3 \times 3$ elliptic systems $\nabla \times (a(x) \nabla\times
u)-\nabla (b(x) \nabla \cdot u)=f$, where the coefficients $a(x)$ and $b(x)$
are positive scalar functions that are measurable and bounded away from zero
and infinity. We prove that weak solutions of the above system are H\"older
continuous under some minimal conditions on the inhomogeneous term $f$. We also
present some applications and discuss several related topics including
estimates of the Green's functions and the heat kernels of the above systems.
