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