Recall that the action of a group $G$ on a probability space $(X,\mathscr{B},\mu)$ is said to be ergodic if, for all $A\in\mathscr{B}$,
$$ \forall g\in G\ \mu(A\Delta g\cdot A)=0 \implies \mu(A)\in\{0,1\}$$
I'm looking for an example of a $\mathbb{Z}^2$ action (say $T$) which is ergodic in the sense above, but such that $T^\vec{n}$ is non-ergodic (in the usual sense) for any $\vec{n}\in\mathbb{Z}^2$.
Intuitively, this would mean that the (almost) invariance under a single $T^{\vec{n}}$ is not sufficient to conclude that the set has null or total measure. However, I don't see how to construct such an action. A hint would be appreciated.
Hint: Take $X = \mathbb{R}^2 / \mathbb{Z}^2$ as the standard $2$-torus with Lebesgue measure, and the action $(a,b) \cdot (x,y) = (x+av, y+bw)$, where $v$ and $w$ are irrational.