Automorphisms of curves $y=y\left(x\right)$, $z=z\left(x\right)$ in ${𝐑}^3$ are investigated; i.e. invertible transformations, where the coordinates of the transformed curve $\overline{y}=\overline{y}\left(\overline{x}\right)$, $\overline{z}=\overline{z}\left(\overline{x}\right)$ depend on the derivatives of the original one up to some finite order $m$. While in the two-dimensional space the problem is completely resolved (the only possible transformations are the well-known contact transformations), the three-dimensional case proves to be much more complicated. Therefore, results (in the form of some systems of partial differential equations...