On measure-preserving selection of solutions of ODEs

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