  We examine the existence of one parameter groups of diffeomorphisms whose
infinitesimal generators annihilate all scalar polynomial curvature invariants
through the application of the Lie derivative, known as
$\mathcal{I}$-preserving diffeomorphisms. Such mappings are a generalization of
isometries and appear to be related to nil-Killing vector fields, for which the
associated Lie derivative of the metric yields a nilpotent rank two tensor. We
show that the set of nil-Killing vector fields contains Lie algebras, although
the Lie algebras may be infinite and can contain elements which are not
$\mathcal{I}$-preserving diffeomorphisms. We then study the curvature structure
of a general Lorenztian manifold, or spacetime, to show that
$\mathcal{I}$-preserving diffeomorphism will only exists for the
$\mathcal{I}$-degenerate spacetimes and to determine when the
$\mathcal{I}$-preserving diffeomorphisms are generated by nil-Killing vector
fields. We identify necessary and sufficient conditions for the degenerate
Kundt spacetimes to admit an additional $\mathcal{I}$-preserving diffeomorphism
and conclude with an application to the class of Kundt spacetimes with constant
scalar polynomial curvature invariants to show that a finite transitive Lie
algebra of nil-Killing vector fields always exists for these spacetimes.
