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We study the most general solution for affine connections that are compatiblewith the variational principle in the Palatini formalism for theEinstein-Hilbert action (with possible minimally coupled matter terms). We findthat there is a family of solutions generalising the Levi-Civita connection,characterised by an arbitrary, non-dynamical vector field ${\cal A}_\mu$. Wediscuss the mathematical properties and the physical implications of thisfamily and argue that, although there is a clear mathematical differencebetween these new Palatini connections and the Levi-Civita one, bothunparametrised geodesics and the Einstein equation are shared by all of them.Moreover, the Palatini connections are characterised precisely by these twoproperties, as well as by other properties of its parallel transport. Based onthis, we conclude that physical effects associated to the choice of one or theother will not be distinguishable, at least not at the level of solutions ortest particle dynamics. We propose a geometrical interpretation for theexistence and unobservability of the new solutions.

On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism