  In recent decades, linear affine threefolds have enabled researchers to solve
some of the challenging problems on affine spaces. Koras-Russell threefolds,
especially the Russell Cubic over $\mathbb{C}$ and Asanuma threefolds over a
field of positive characteristic, are striking examples of such linear
threefolds.In this paper, we apply tools from $K$-theory and theory of
$\mathbb{G}_a$-actions to linear threefolds of the form $G:=a(X)Y-F(X,Z,T)\in
k[X,Y,Z,T]$, over an arbitrary field $k$. We give some equivalent conditions
for $G$ to be a hyperplane (i.e., $k[X,Y,Z,T]/(G)=k^{[3]}$) in the following
cases: (i) $k$ is a field of characteristic zero (ii) $k$ is an arbitrary field
and $a(X)$ has only multiple roots. We also establish the Abhyankar-Sathaye
Conjecture affirmatively in these cases.
