  For every $k \in \mathbb{N}$ and $\alpha \in (0,1)$ we construct a
divergence-free $u \in C^k([0,T],C^\alpha(\mathbb{T}^d,\mathbb{R}^d))$, $d \geq
2$, such that there is no measurable selection of solutions of the ODE
$\dot{X}_t = u(t,X_t)$ that preserves the Lebesgue measure.
