  In this work, we investigate a theory of linear Weyl gravity coupled to a
scalar field and study the scenario in which Lorentz symmetry is broken by a
non-vanishing vacuum expectation value of the Weyl field in the flat space
limit after Weyl symmetry breaking. We show that a $CPT$-odd Lorentz-violating
interaction is generated after symmetry breaking. Features of different
symmetry-broken phases and their dependence on the spacetime character of the
generated Lorentz-violating background are discussed. Also, we analyze the
naturalness of the theory by showing that the light mass scale is protected
from large radiative corrections due to an enhanced spacetime symmetry.
