  Working directly with a general Hamiltonian for the spacetime metric with the
$3+1$ decomposition and keeping only the spatial covariance, we investigate the
possibility of reducing the number of degrees of freedom by introducing an
auxiliary constraint. The auxiliary constraint is considered as part of the
definition of the theory. Through a general Hamiltonian analysis, we find the
conditions for the Hamiltonian as well as for the auxiliary constraint, under
which the theory propagates two tensorial degrees of freedom only. The class of
theories satisfying these conditions can be viewed as a new construction for
the type-II minimally modified gravity theories, which propagate the same
degrees of freedom of but are not equivalent to general relativity in the
vacuum. We also illustrate our formalism by a concrete example, and derive the
dispersion relation for the gravitational waves, which can be constrained by
observations.

General Relativity and Quantum Cosmology;High Energy Physics - Theory