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 α the solubility problem is studied for the additive cohomological equation on the torus w(Tαx)-w(x)=f(x)-∫Tdf(t) dt, where Tαx=x+α (mod 1) the shift of the torus Td by the vector α 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 x-dot=α/(F(x,y)), y-dot=1/(F(x,y)) where F(x,y) is a positive 1-periodic function of finite smoothness.