  We investigate the possibility of reducing the number of degrees of freedom
(d.o.f.) starting from generic metric theories of gravity by introducing
multiple auxiliary constraints (ACs), under the restriction of retaining
spatial covariance as a gauge symmetry. Arbitrary numbers of scalar-, vector-
and tensor-type ACs are considered a priori, yet we find that no vector- and
tensor-type constraints should be introduced, and that scalar-type ACs should
be no more than four for the purpose of constructing minimally modified gravity
(MMG) theories which propagate only two tensorial d.o.f., like general
relativity (GR). Through a detailed Hamiltonian analysis, we exhaust all the
possible classifications of ACs and find out the corresponding minimalizing and
symmetrizing conditions for obtaining the MMG theories. In particular, no
condition is required in the case of four ACs, hence in this case the theory
can couple with matter consistently and naturally. To illustrate our formalism,
we build a concrete model for this specific case by using the Cayley-Hamilton
theorem and derive the dispersion relation of the gravitational waves, which is
subject to constraints from the observations.
