  In this paper we propose a wider class of symmetries including the Galilean
shift symmetry as a subclass. We will show how to construct ghost-free nonlocal
actions, consisting of infinite derivative operators, which are invariant under
such symmetries, but whose functional form is not simply given by exponentials
of entire functions. Motivated by this, we will consider the case of a scalar
field and discuss the pole structure of the propagator which has infinitely
many complex conjugate poles, but satisfies the tree-level unitarity. We will
also consider the possibility to construct UV complete Galilean theories by
showing how the ultraviolet behavior of loop integrals can be ameliorated.
Moreover, we will consider kinetic operators respecting the same symmetries in
the context of linearized gravity. In such a scenario, the graviton propagator
turns out to be ghost-free and the spacetime metric generated by a point-like
source is nonsingular. These new nonlocal models can be seen as an infinite
derivative generalization of Lee-Wick theories and open a new branch of
nonlocal theories.
