  We present a new automated method for finding integrable symplectic maps of
the plane. These dynamical systems possess a hidden symmetry associated with an
existence of conserved quantities, i.e. integrals of motion. The core idea of
the algorithm is based on the knowledge that the evolution of an integrable
system in the phase space is restricted to a lower-dimensional submanifold.
Limiting ourselves to polygon invariants of motion, we analyze the shape of
individual trajectories thus successfully distinguishing integrable motion from
chaotic cases. For example, our method rediscovers some of the famous
McMillan-Suris integrable mappings and discrete Painlev\'e equations. In total,
over 100 new integrable families are presented and analyzed; some of them are
isolated in the space of parameters, and some of them are families with one
parameter (or the ratio of parameters) being continuous or discrete. At the end
of the paper, we suggest how newly discovered maps are related to a general 2D
symplectic map via an introduction of discrete perturbation theory and propose
a method on how to construct smooth near-integrable dynamical systems based on
mappings with polygon invariants.
