  The nonlinear stability domain of Lagrange's celebrated 1772 solution of a
three-body problem is obtained numerically as a function of the masses of the
bodies and the common eccentricity of their Keplerian orbits. This domain shows
that this solution may be realized in extra-solar planetary systems similar to
those that have been discovered recently with two Jupiter-size planets orbiting
a solar-size star. For an exact 1:1 resonance, the Doppler shift variation in
the emitted light would be the same as for stars which have only a single
planetary companion. But it is more likely that in actual extra-solar planetary
systems there are deviations from such a resonance, raising the interesting
prospect that Lagrange's solution can be identified by an analysis of the
observations. The existence of another stable 1:1 resonance solution which
would have a more unambiguous Doppler shift signature is also discussed.
