  We pursue the study of one-dimensional symmetry of solutions to nonlinear
equations involving nonlocal operators. We consider a vast class of nonlinear
operators and in a particular case it covers the fractional $p-$Laplacian
operator. Just like the classical De Giorgi's conjecture, we establish a
Poincar\'e inequality and a linear Liouville theorem to provide two different
proofs of the one-dimensional symmetry results in two dimensions. Both
approaches are of independent interests. In addition, we provide certain energy
estimates for layer solutions and Liouville theorems for stable solutions. Most
of the methods and ideas applied in the current article are applicable to
nonlocal operators with general kernels where the famous extension problem,
given by Caffarelli and Silvestre, is not necessarily known.
