Let $(X, d)$ be a compact metric space, let $T$ be a group of actions on $X$. Then $(X,T)$ is a topological dynamical system with transformation group $T$, and we denote it by $(X,T)$. We say points $x, y\in X$ are proximal when we have that $$\inf_{t\in T}\ d(tx, ty)=0.$$ A point $x\in X$ is called distal if it is only proximal to itself.

A topological dynamical system $(X,T)$ is called point-distal if there exists a point $x\in X$ such that $x$ is distal and the orbit of $x$ is dense in $X$.

Minimal systems which are not point-distal are not hard to be produced. But when $T$ is abelian, there does not seem to be a large store of examples, according to Veech's paper "point-distal flow"(MR0267560). In this paper, Veech gave a example of a type of minimal systems which are not point-distal: the horocycle flows. My question is that: when $T=\mathbb{Z}$, is there any other example of this kind of dynamical system? I think such system exists, but I don't know how to construct one.

Let $(X, d)$ be a compact metric space, let $T$ be a group of actions on $X$. Then $(X,T)$ is a topological dynamical system with transformation group $T$, and we denote it by $(X,T)$. We say points $x, y\in X$ are proximal when we have that $$\inf_{t\in T}\ d(tx, ty)=0.$$ A point $x\in X$ is called distal if it is only proximal to itself.

A topological dynamical system $(X,T)$ is called point-distal if there exists a point $x\in X$ such that $x$ is distal and the orbit of $x$ is dense in $X$.

Minimal systems which are not point-distal are not hard to be produced. But when $T$ is abelian, there does not seem to be a large store of examples, according to Veech's paper "point-distal flow"(MR0267560). In this paper, Veech gave a example of a type of minimal systems which are not point-distal: the horocycle flows. 

My question is that: when $T=\mathbb{Z}$, except the horocycle flows, is there any other example of minimal dynamical system which is not point distal? I think such system exists, but I don't know how to construct one.