  New electromagnetic conservation laws have recently been proposed: in the
absence of electromagnetic currents, the trace of the Chevreton superenergy
tensor, $H_{ab}$ is divergence-free in four-dimensional (a) Einstein spacetimes
for test fields, (b) Einstein-Maxwell spacetimes. Subsequently it has been
pointed out, in analogy with flat spaces, that for Einstein spacetimes the
trace of the Chevreton superenergy tensor $H_{ab}$ can be rearranged in the
form of a generalised wave operator $\square_L$ acting on the energy momentum
tensor $T_{ab}$ of the test fields, i.e., $H_{ab}=\square_LT_{ab}/2$. In this
letter we show, for Einstein-Maxwell spacetimes in the full non-linear theory,
that, although, the trace of the Chevreton superenergy tensor $H_{ab}$ can
again be rearranged in the form of a generalised wave operator $\square_G$
acting on the electromagnetic energy momentum tensor, in this case the result
is also crucially dependent on Einstein's equations; hence we argue that the
divergence-free property of the tensor $H_{ab}=\square_GT_{ab}/2$ has
significant independent content beyond that of the divergence-free property of
$T_{ab}$.
