We consider rotated $n$-Laplace systems on the unit ball $B_1 \subset \mathbb{R}^n$ of the form \begin{align*} -\mathrm{div}\left( Q|\nabla u|^{n-2} \nabla u\right) = \mathrm{div}(G), \end{align*} where $u\in W^{1,n}(B_1;\mathbb{R}^N)$, $Q\in W^{1,n}(B_1;SO(N))$ and $G\in L^{\left( \frac{n}{n-1},q \right)}(B_1;\mathbb{R}^n\otimes \mathbb{R}^N)$ for some $0Comment: 14 pages. Added in v2: the constant $\theta$ in Theorem 1.1 has been improved
