Let $v:\mathbb R^3 \to \mathbb R^3$ a vector field such that $\forall x\in\mathbb R^3, (\nabla \times v) (x) = \rho~$ where $\rho$ is a constant. Example here.
Can you characterize $v$ simply?
What I have so far:
If I'm not mistaken, a vector field of the form $x \mapsto a + b \times x$ has a curl of $2b$ (with $a,b\in\mathbb R^3$ and $\times$ the cross-product). Is this the only form of vector field with uniform curl?
As far as I understand the curl of $v$ being uniform is equivalent to the antisymmetric part of the Jacobian of $v$ being uniform. If the symmetric part of the Jacobian of $v$ is zero, then we would thus have $J_v(x)x = Bx = b\times x$ for some antisymmetric matrix $B$, which might be part of the answer.
Is there a reason to think the symmetric part is equal to zero? This would mean that $\nabla \cdot v=0$ right?
No, you could add to your $v$ any irrotational vector field, i.e. $w$ with $\nabla \times w = 0$. In particular, $w$ could be the gradient of any twice differentiable function, since $\nabla \times \nabla f = 0$.