  The Standard-Model Extension provides a framework to systematically
investigate possible violation of the Lorentz symmetry. Concerning gravity, the
linearized version was extensively examined. We here cast the first set of
experimental bounds on the nonlinear terms in the field equation from the
anisotropic cubic curvature couplings. These terms introduce body-dependent
accelerations for self-gravitating objects, thus violating the gravitational
weak equivalence principle (GWEP). Novel phenomena, that are absent in the
linearized gravity, remain experimentally unexplored. We constrain them with
precise binary-orbit measurements from pulsar timing, wherein the high density
and large compactness of neutron stars are crucial for the test. It is the
first study that seeks GWEP-violating signals in a fully anisotropic framework
with Lorentz violation.
