  Many advancements have been made in the field of topological mechanics. The
majority of the works, however, concerns the topological invariant in a linear
theory. We, in this work, present a generic prescription of defining
topological indices which accommodates non-linear effects in mechanical systems
without taking any approximation. Invoking the tools of differential geometry,
a Z-valued quantity in terms of the Poincare-Hopf index, that features the
topological invariant of non-linear zero modes (ZMs), is predicted. We further
identify one type of topologically protected solitons that are robust to
disorders. Our prescription constitutes a new direction of searching for novel
topologically protected non-linear ZMs in the future.
