Suppose that $G$ be the Lie symmetry group for a given set of differential equations (DEs) $\mathcal{D}$. Let $\mathcal{N}$ be the manifold representing $\mathcal{D}$ in jet space $\mathcal{P}$; i.e. $\mathcal{N} \subset \mathcal{P}$ is an embedding. Then, of course, $g.\mathcal{N} \subseteq \mathcal{N}$, $\forall\; g \in G$. Furthermore, any $g.[x, u]=[\bar{x}, \bar{u}]$ is the graph of a new solution $\bar{u}(\bar{x})$, as expected, where $x$ and $u$ stand for independent and dependent variables respectively.
The question is “whether the same G could be utilized to translate a solution $u(x)$ along itself”; i.e.: $$ g.[x, u] = [ax-t, u(ax-t)], $$ where $a,t \in \mathbb{R}$; s.t. $g \in G$ exists? By the way, it is , of course, valid for partial differential equations as well. My notation is just for the sake of clarity.