  In this work, we prove a new decomposition result for rank $m$ symmetric
tensor fields which generalizes the well known solenoidal and potential
decomposition of tensor fields. This decomposition is then used to describe the
kernel and to prove an injectivity result for first $(k+1)$ integral moment
transforms of symmetric $m$-tensor fields in $\mathbb{R}^n$. Additionally, we
also present a range characterization for first $(k+1)$ integral moment
transforms in terms of the John's equation.
