  Recently a new class of critical points, termed as {\sl perpetual points},
where acceleration becomes zero but the velocity remains non-zero, is observed
in nonlinear dynamical systems. In this work we show whether a transformation
also maps the perpetual points to another system or not. We establish
mathematically that a linearly transformed system is topologicaly conjugate,
and hence does map the perpetual points. However, for a nonlinear
transformation, various other possibilities are also discussed. It is noticed
that under a linear diffeomorphic transformation, perpetual points are mapped,
and accordingly, eigenvalues are preserved.
