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We derive all the reductions of the system of two coupled sine-Gordonequations introduced by Konopelchenko and Rogers to ordinary differentialequations. All these reductions are degeneracies of a master reduction to anequation found by Chazy "curious for its elegance", an algebraic transform ofthe most general sixth equation of Painlev\'e. -- --  Nous \'etablissons toutes les r\'eductions du syst\`eme de deux \'equationscoupl\'ees de sine-Gordon introduit par Konopelchenko et Rogers \`a des\'equations diff\'erentielles ordinaires. Ces r\'eductions sont toutes desd\'eg\'en\'erescences d'une r\'eduction ma{\^\i}tresse \`a une \'equationjug\'ee par Chazy "curieuse en raison de [son] \'el\'egance", transform\'eealg\'ebrique de la sixi\`eme \'equation de Painlev\'e la plus g\'en\'erale.

R\'eductions d'un syst\`eme bidimensionnel de sine-Gordon \`a la sixi\`eme \'equation de Painlev\'e