  We attempt to generalize the familiar covariantly conserved Bel-Robinson
tensor B_{mnab} ~ R R of GR and its recent topologically massive third
derivative order counterpart B ~ RDR, to quadratic curvature actions. Two very
different models of current interest are examined: fourth order D=3 "new
massive", and second order D>4 Lanczos-Lovelock, gravity. On dimensional
grounds, the candidates here become B ~ DRDR+RRR. For the D=3 model, there
indeed exist conserved B ~ dRdR in the linearized limit. However, despite a
plethora of available cubic terms, B cannot be extended to the full theory. The
D>4 models are not even linearizable about flat space, since their field
equations are quadratic in curvature; they also have no viable B, a fact that
persists even if one includes cosmological or Einstein terms to allow
linearization about the resulting dS vacua. These results are an unexpected, if
hardly unique, example of linearization instability.
