2.1.I * Let the set $M$ be the collection of all functions $f(x)$ that satisfy the following property: there exists a non-zero constant $T$, such that for any $x \in \mathbf{R}$, $f(x+T)=T f(x)$ holds. If the function $f(x)=\sin k x \in M$, then the range of the real number $k$ is $\qquad$ .
When $k=0$, $f(x)=0 \in M$.
When $k \neq 0$, since $f(x)=\sin k x \in M$, there exists a non-zero constant $T$, for any $x \in \mathbf{R}$, such that $\sin (k x+k T)=T \sin k x$.

When $x=0$, $\sin k T=0$. Therefore, for any $x \in \mathbf{R}$, $\cos k T \cdot \sin k x=T \sin k x$. Hence $T=\cos k T= \pm 1$. Substituting $\sin k T=0$, we get $k=m \pi, m \in \mathbf{Z}$