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We analyse the most general connection allowed by Einstein-Hilbert theory inPalatini formalism. We also consider a matter lagrangian independent of theaffine connection. We show that any solution of the equation of the connectionis essentially Levi-Civita up to a term that contains an undetermined 1-form.Finally, it is proved that these connections and Levi-Civita describe acompletely equivalent physics.

(Non-)uniqueness of Einstein-Palatini Gravity