  We investigate a class of Lagrangians that admit a type of perturbed harmonic
oscillator which occupies a special place in the literature surrounding
perturbation theory. We establish explicit and generalized geometric conditions
for the symmetry determining equations. The explicit scheme provided can be
followed and specialized for any concrete perturbed differential equation
possessing the Lagrangian. A systematic solution of the conditions generate
nontrivial approximate symmetries and transformations. Detailed cases are
discussed to illustrate the relevance of the conditions, namely (a) $G_1$ as a
quadratic polynomial, (b) the Klein-Gordon equation of a particle in the
context of Generalized Uncertainty Principle and (c) an orbital equation from
an embedded Reissner-Nordstr\"om black hole.
