  A polynomial automorphism $F$ is called {\em shifted linearizable} if there
exists a linear map $L$ such that $LF$ is linearizable. We prove that the
Nagata automorphism $N:=(X-Y\Delta -Z\Delta^2,Y+Z\Delta, Z)$ where
$\Delta=XZ+Y^2$ is shifted linearizable. More precisely, defining $L_{(a,b,c)}$
as the diagonal linear map having $a,b,c$ on its diagonal, we prove that if
$ac=b^2$, then $L_{(a,b,c)}N$ is linearizable if and only if $bc\not = 1$. We
do this as part of a significantly larger theory: for example, any exponent of
a homogeneous locally finite derivation is shifted linearizable. We pose the
conjecture that the group generated by the linearizable automorphisms may
generate the group of automorphisms, and explain why this is a natural
question.
