  In this work, we investigate the system formed by the equations $\text{div }
\vec w=g_0$ and $\text{curl } \vec w=\vec g$ in bounded star-shaped domains of
$\mathbb{R}^3$. A Helmholtz-type decomposition theorem is established based on
a general solution of the above-mentioned div-curl system which was previously
derived in the literature. When $g_0\equiv 0$, we readily obtain a bounded
right inverse of $\text{curl}$ which is a divergence-free invariant. The
restriction of this operator to the subspace of divergence-free vector fields
with vanishing normal trace is the well-known Biot--Savart operator. In turn,
this right inverse of $\text{curl}$ will be modified to guarantee its
compactness and satisfy suitable boundary-value problems. Applications to
Beltrami fields, Vekua-type problems as well as Maxwell's equations in
inhomogeneous media are included.
