Rigidity of a class of smooth singular flows on ${\mathbb{T}}^2$

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field $\mathcal{X}$ on $\mathbb T^2\setminus {a}$,where$$\mathcal{X}$$isnotdefinedat$a\in \mathbb T^2$. It follows that the phase space can be decomposed into a (topological disc) $D_{\mathcal{X}}$ and an ergodic component $E_$\mathcal{X}$=\mathbb T^2\setminus D_$\mathcal{X}$$\mathrm{.}Let$\omega_$\mathcal{X}$$bethe1-formassociatedto$$\mathcal{X}$$\mathrm{.}Weshowthatif$|\int_{E_{$\mathcal{X}$1}}d\omega{$\mathcal{X}$1}|\neq |\int{E_{$\mathcal{X}$2}}d\omega{$\mathcal{X}$_2}|$, then the corresponding flows $(v_t^{{\mathcal{X}}_1})$ and $(v_t^{{\mathcal{X}}_2})$ are disjoint. It also follows that for every $\mathcal{X}$ there is a uniquely associated frequency $\alpha ={\alpha }_{\mathcal{X}}\in \mathbb{T}$. We show that for a full measure set of $\alpha \in \mathbb{T}$ the class of smooth time changes of $(v_t^{{\mathcal{X}}_{\mathcal{\alpha }}})$ is joining rigid, i.e. every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to a problem of Ratner (Problem 3 in \cite{Rat4}) is positive.