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It is proven that second-order vectorial nonlinear differential systemsy''=f(y) , possess a continuum of symmetric solutions. They are shown topossess a continuum of even solutions. If f(y) is an odd function of y , theny''=f(y) is shown also to possess a continuum of odd solutions. The resultsapply to a significant family of second-order vectorial nonlinear differentialsystems that are not dissipative. This family of differential equationsincludes the celebrated N-body problem of celestial mechanics and other centralforce problems.

Symmetric Solutions to Nonlinear Vectorial 2nd Order ODE's