Let $E$ be an elliptic curve given in long Weierstraß form with all coefficients $a_1,a_2,a_3,a_4,a_6 \in \mathbb{Z}$. It is known that the rational points $E(\mathbb{Q})$ form a group which has a finite torsion subgroup $T$.

- Question:

What is known about the action of the following subgroup $\hat{T} \le T$ on the integral points $E(\mathbb{Z})$:

$$\hat{T} := \{ t \in T | t + E(\mathbb{Z}) \subset E(\mathbb{Z}) \}$$

?

I have found one elliptic curve where this $\hat{T}$ is not the trivial group:

https://www.lmfdb.org/EllipticCurve/Q/210/e/6

For this curve:

$$T := \left[\left(0 : 1 : 0\right), \left(4 : 58 : 1\right), \left(64 : 418 : 1\right), \left(-26 : 148 : 1\right), \left(28 : -14 : 1\right), \left(-26 : -122 : 1\right), \left(64 : -482 : 1\right), \left(4 : -62 : 1\right), \left(-36 : 18 : 1\right), \left(34 : -122 : 1\right), \left(-8 : -122 : 1\right), \left(244 : -3902 : 1\right), \left(\frac{31}{4} : -\frac{31}{8} : 1\right), \left(244 : 3658 : 1\right), \left(-8 : 130 : 1\right), \left(34 : 88 : 1\right)\right] $$

and

$$\hat{T}:= \left[\left(0 : 1 : 0\right), \left(\frac{31}{4} : -\frac{31}{8} : 1\right)\right]$$

I must admit, that in most cases where I looked at numerical examples, we had $\hat{T} = 1$.

- Question: Are there examples of elliptic curves with $1 < \hat{T} = T$?

Thanks for your help.