Recurrence of the integral of a smooth three-frequency conditionally periodic function

We prove V. V. Kozlov's famous conjecture claiming that the integral of an analytic three-frequency conditionally periodic function with zero mean and incommensurable frequencies recurs. For a conditionally periodic function of class $C^2$ on $\mathbb T^n$, $n=2,3$, we prove that the integral recurs uniformly with respect to the initial data.