  The aim of this paper is to find higher order geometrical corrections to the
Einstein-Hilbert action that can lead to only second order equations of motion.
The metric formalism is used, and static spherically symmetric and
Friedmann-Lema\^itre space-times are considered, in four dimensions. The
FKWC-basis are introduced in order to consider all the possible invariant
scalars, and both polynomial and non-polynomial gravities are investigated.
