Is there a non-affine harmonic map $\mathbb{R}^3 \to \mathbb{R}^3$ with constant Jacobian?

Question

Does there exist a smooth harmonic map $f:\mathbb{R}^3 \to \mathbb{R}^3$ such that:

1. $\det(df)$ is constant and nonzero.
2. $f$ is not affine.

($f$ is harmonic if each of its three components is a harmonic function).

The case of maps $\mathbb{R}^2 \to \mathbb{R}^2$ has been settled here. (There is no such map). However, the proof there used techniques from complex analysis, which are not available in dimension $3$.

Answer 1

Contrary to the two-dimensional case, there exist such maps. For instance, take any non-affine harmonic map $u:\mathbb{R}^2 \to \mathbb{R}$ and define $f:\mathbb{R}^3 \to \mathbb{R}^3$ by $f(x,y,z) = ( x + u(y,z), y , z)$. Then $f$ is harmonic, non-affine and $\det df =1$.