  In this paper, we explore various equivalences of Ekeland's variational
principle within the framework of group-invariant mappings. We introduce and
analyze several key theorems, including the Drop theorem, the Petal theorem,
Caristi-Kirk fxed-point theorem, and Takahashi's theorem, all of them within
this context. Moreover, we extend the classical Drop theorem and Petal theorem
to a more generalized setting. We also demonstrate the practical signifcance of
these findings through numerous applications to diverse areas of mathematics.
In particular, in the context of partial differential equations, we explore
their implications on the solution of the Plateau problem, and in control
theory. We also extend the classical Pontyargin maximum principle.
