Extension of the parametrized post-Newtonian formalism to the case of a quasi-Lorentz frame of reference

It is shown that using of the post-Newtonian metric in the canonical frame of reference is physically incorrect in some cases. Principal relationships of the post-Newtonian formalism for the metric-based theories of gravity are considered in the quasi-Lorentz frame of reference with the introduction of a new parameter. Conditions of the existence of conservation laws are obtained.