  We construct explicit examples of deformed $G_2$-instantons, also called
Donaldson-Thomas connections, on $\mathbb{R}^4 \times S^3$ endowed with the
torsion free $G_2$-structure found by Brandhuber et al. and on
$\mathbb{R}^+\times S^3 \times S^3$ endowed with the Bryant-Salamon conical
$G_2$-structure. These are the first such non-trivial examples on a $G_2$
manifold. As a by-product of our investigation we also find an associative
foliation of $\mathbb{R}^4\times S^3$ by $\mathbb{R}^2 \times S^1$.

Differential Geometry