  Algebraic actions of unipotent groups $U$ actions on affine $k-$varieties $X$
($k$ an algebraically closed field of characteristic 0) for which the algebraic
quotient $X//U$ has small dimension are considered$.$ In case $X$ is factorial,
$O(X)^{\ast}=k^{\ast},$ and $X//U$ is one-dimensional, it is shown that
$O(X)^{U}$=$k[f]$, and if some point in $X$ has trivial isotropy, then $X$ is
$U$ equivariantly isomorphic to $U\times A^{1}(k).$ The main results are given
distinct geometric and algebraic proofs. Links to the Abhyankar-Sathaye
conjecture and a new equivalent formulation of the Sathaye conjecture are made.
