  We argue that there is nothing puzzling in the fact that the Hamiltonian
formulation of a covariant theory, General Relativity, after a non-covariant
change of field variables is not canonically related to the formulation based
on the original variable, the metric tensor. Were such a puzzle to be "solved"
it would lead to the conclusion that a covariant theory can be converted into a
non-covariant one in many different ways and without consequence. The
non-canonicity of transformations from covariant to non-covariant variables
shows the need to work in the original variables so as to be able to restore
the covariant gauge transformations in the Hamiltonian approach. Any
modification of Dirac's procedure for the constrained Hamiltonian with the aim
to prove the legitimacy of non-covariant changes of field variables, or
rejection of Dirac's procedure as "not fundamental and undoubted" because it
does not allow such changes (as recently suggested by Shestakova [CGQ, 28
(2011) 055009]), is equivalent to the rejection of the covariance of General
Relativity, and to surrender to such temptation is truly puzzling.
