  We prove that if $P(D)$ is some constant coefficient partial differential
operator and $f$ is a scalar field such that $P(D)f$ vanishes in a given open
set, then the integrals of $f$ over all lines intersecting that open set
determine the scalar field uniquely everywhere. This is done by proving a
unique continuation property of fractional Laplacians which implies uniqueness
for the partial data problem. We also apply our results to partial data
problems of vector fields.
