  We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine
Lovelock theories. First we study the projective invariance of general Lovelock
actions and show that all connections constructed by acting with a projective
transformation of the Levi-Civita connection are allowed solutions, albeit
physically equivalent to Levi-Civita. We then show that the (non-integrable)
Weyl connection is also a solution for the specific case of the
four-dimensional metric-affine Gauss-Bonnet action, for arbritrary vector
fields. The existence of this solution is related to a two-vector family of
transformations, that leaves the Gauss-Bonnet action invariant when acting on
metric-compatible connections. We argue that this solution is physically
inequivalent to the Levi-Civita connection, giving thus a counterexample to the
statement that the metric and the Palatini formalisms are equivalent for
Lovelock gravities. We discuss the mathematical structure of the set of
solutions within the space of connections.
