  Theory known as Special Galileon has recently attracted considerable interest
due to its peculiar properties. It has been shown that it represents an
extremal member of the set of effective field theories with enhanced soft
limit. This property makes its tree-level S-matrix fully on-shell
reconstructible and representable by means of the Cachazo-He-Yuan
representation. The enhanced soft limit is a consequence of new hidden symmetry
of the Special Galileon action, however, until now, the origin of this peculiar
symmetry has remained unclear. In this paper we interpret this symmetry as a
special transformation of the coset space $GAL(D,1)/SO(1,D-1)$ and show, that
there exists a three-parametric family of invariant Galileon actions. The
latter family is closed under duality which appears as a natural generalization
of the above mentioned symmetry. We also present a geometric construction of
the Special Galileon action using $D$-dimensional brane propagating in
$2D$-dimensional flat pseudo-riemannian space. Within such framework, the
Special Galileon symmetry emerges as an $U(1,D-1)$ symmetry of the target
space, which can be treated as a $D$-dimensional K\"ahler manifold. Such a
treatment allows for classification of the higher order invariant Lagrangians
needed as counterterms on the quantum level. We also briefly comment on
relation between such higher order Lagrangians and the Lagrangians invariant
with respect to the polynomial shift symmetry.
