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We prove that all non-degenerate relative equilibria of the planar Newtonian$n$--body problem can be continued to spaces of constant curvature $\kappa$,positive or negative, for small enough values of this parameter. We alsocompute the extension of some classical relative equilibria to curved spacesusing numerical continuation. In particular, we extend Lagrange's triangleconfiguration with different masses to both positive and negative curvaturespaces.

Continuation of relative equilibria in the $n$--body problem to spaces of constant curvature