  Let $I_{m}$ denote the Euclidean ray transform acting on compactly supported
symmetric $m$-tensor field distributions $f$, and $I_{m}^{*}$ be its formal
$L^2$ adjoint. We study a unique continuation result for the normal operator
$N_{m}=I_{m}^{*}I_{m}$. More precisely, we show that if $N_{m}$ vanishes to
infinite order at a point $x_0$ and if the Saint-Venant operator $W$ acting on
$f$ vanishes on an open set containing $x_0$, then $f$ is a potential tensor
field. This generalizes two recent works of Ilmavirta and M\"onkk\"onen who
proved such unique continuation results for the ray transform of functions and
vector fields/1-forms. One of the main contributions of this work is
identifying the Saint-Venant operator acting on higher order tensor fields as
the right generalization of the exterior derivative operator acting on 1-forms,
which makes unique continuation results for ray transforms of higher order
tensor fields possible. In the second half of the paper, we prove analogous
unique continuation results for momentum ray and transverse ray transforms.
