On the additive cohomological equation and time change for a linear flow on the torus with a Diophantine frequency vector

For a 1-periodic function $f$ of finite smoothness and a Diophantine vector $\alpha$ the solubility problem is studied for the additive cohomological equation on the torus
$$ w(T_\alpha x)-w(x)=f(x)-\int_{\mathbb T^d}f(t)\,dt, $$
where $T_\alpha x=x+\alpha\pmod1$ is the shift of the torus $\mathbb T^d$ by the vector $\alpha$ and $w$ is an unknown measurable function.
Necessary and sufficient conditions are obtained for the conjugacy of a linear flow on the $(d+1)$-torus to the reparametrized flow
$$ \begin{cases} \dot x=\dfrac\alpha{F(x,y)}\,,\\ \dot y=\dfrac1{F(x,y)}\,, \end{cases} $$
where $F(x,y)$ is a positive 1-periodic function of finite smoothness.