  We establish arithmeticity in the sense of A. Katok and F. Rodriguez Hertz of
smooth actions $\alpha$ of $\mathbb{R}^k$ on an anonymous manifold $M$ of
dimension $2k+1$ provided that there is an ergodic invariant Borel probability
measure on $M$ w/r/t which each nontrivial time-$t$ map $\alpha_t$ of the
action has positive entropy. Arithmeticity in this context means that the
action $\alpha$ is measure theoretically isomorphic to a constant time change
of the suspension of an affine Cartan action of $\mathbb{Z}^k$. This in
particular solves, up to measure theoretical isomorphism, Problem 4 from a
prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin.
