An integrability illusion: Vanishing transfer matrices associated with generalised Gaudin superalgebras

The Lie superalgebra $gl\left(m\right|n)$ admits irreducible, finite-dimensional representations that continuously depend on a free parameter. For each representation, we define a generalised Gaudin superalgebra through an associated solution of the classical Yang-Baxter equation. The universal enveloping superalgebra of the Gaudin superalgebra formally contains a commuting family of transfer matrices, given by a universal expression. It will be shown that in many instances these transfer matrices are identically zero in the universal enveloping superalgebra. We offer an alternative formulation for identifying transfer matrices.