  We study Birkhoff averages along trajectories of smooth reparameterizations
of irrational linear flows of the two torus with two stopping points, say
$\mathbf p$ and $\mathbf q$, of quadratic order. The limiting behaviour of such
averages is independent of the starting point in a set of full Haar-Lebesgue
measure and depends in an intricate way on the Diophantine properties of both
the slope $\alpha$ of the linear flow as well as the relative position of
$\mathbf p$ and $\mathbf q$. In particular, if $\alpha$ is Diophantine, then
Birkhoff limits diverge almost everywhere (historic behaviour) and if $\alpha$
is sufficiently Liouville, then there exists some $\mathbf p$ and $\mathbf q$
such that the Birkhoff averages converge almost everywhere (unique physical
measure).
