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automorphisms of curves; infinite-dimensional space; contact forms
Automorphisms of curves $y= y(x)$, $z=z(x)$ in ${\bold R}^3$ are investigated; i.e. invertible transformations, where the coordinates of the transformed curve $\bar y=\bar y(\bar x)$, $\bar z= \bar z(\bar x)$ 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 for the functions, determining the automorphisms) only for the special case $\bar x =x$ and order $m\leq 2$ are obtained. Finally, the problem of infinitesimal transformations is briefly mentioned.
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