  We construct a nonlinear lattice that has a particular symmetry in its
potential function consisting of long-range pairwise interactions. The symmetry
enhances smooth propagation of discrete breathers, and it is defined by an
invariance of the potential function with respect to a map acting on the
complex normal mode coordinates. Condition of the symmetry is given by a set of
algebraic equations with respect to coefficients of the pairwise interactions.
We prove that the set of algebraic equations has a unique solution, and
moreover we solve it explicitly. We present an explicit Hamiltonian for the
symmetric lattice, which has coefficients given by the solution. We demonstrate
that the present symmetric lattice is useful for numerically computing
traveling discrete breathers in various lattices. We propose an algorithm using
it.
