It is known that a part from the zero tensor field, we can only recover the solenoidal part of a tensor field field of arbitrary order from its X-ray transform.
Given the vector field $F = (f_1,f_2),$ and $\theta =(\cos\varphi,\sin\varphi),$ the X-ray transform of $F$ is given by
$$\displaystyle\cos\varphi \int_{-\infty}^\infty f_1(x+t\theta)dt + \sin\varphi \int_{-\infty}^\infty f_2(x+t\theta)dt = g(x,\theta). $$
It is easy to see that this has the unique solution $$ \int_{-\infty}^\infty f_1(x+t\theta)dt =\cos\varphi g(x,\theta),\quad \mbox{and} \quad \int_{-\infty}^\infty f_2(x+t\theta)dt =\sin\varphi g(x,\theta). $$
These are the X-ray transform of scalar functions which are invertible, and thus the vector field can be uniquely recovered from its X-ray transform. What went wrong?. Thanks