  In this article generalized attenuated ray transforms (ART) and integral
angular moments are investigated. Starting from the Radon transform, the
attenuated ray transform and the longitudinal ray transform, we derive the
concept of ART-operators of order $k$ over functions defined on the phase space
and depending on time. The ART-operators are generalized for complex-valued
absorption coefficient as well as weight functions of polynomial and
exponential type. Connections between ART operators of various orders are
established by means of the application of the linear part of a transport
equation. These connections lead to inhomogeneous differential equations of
order $(k+1)$ for the ART of order $k$. Uniqueness theorems for the
corresponding boundary-value and initial boundary-value problems are proved.
Properties of integral angular moments of order $p$ are considered and
connections between the moments of different orders are deduced. A close
connection of the considered operators with mathematical models for tomography,
physical optics and integral geometry allows to treat the inversion of ART of
order $k$ as an inverse problem of determining the right-hand side of a
corresponding differential equation.
