Is there an "interesting" class of analytically-integrable, divergence-free vector fields over $\mathbb{R}^2$ and/or $\mathbb{R}^3$?

That is, I'm looking for a large class of vector fields given by $V(x;\eta)$ for $x\in\mathbb{R}^k$, $k\in\{2,3\}$, parameterized by a fixed set of values $\eta\in\mathbb{R}^n$ such that:

- $\nabla_x \cdot V(x;\eta)=0\ \forall x\in\mathbb{R}^k$
- There exists an easily-evaluated (closed-form?) function $\Phi(t;x_0,\eta)$ such that the ODE $p'(t)=V(p(t);\eta)$ is solved by taking $p(t)=\Phi(t;p(0),\eta).$

For example, the vector fields $V(x;A)=Ax$ admit $\Phi(t;p_0,\eta)=e^{At}p_0$, but the subset of $A$'s giving divergence-free vector fields is not too rich/interesting.