Nonprojectable Horava-Lifshitz gravity without the unwanted scalar graviton

We consider a way of eliminating the unwanted scalar graviton from Horava-Lifshitz gravity that is achieved via introduction of certain additional constraints. We perform canonical analysis of both projectable and nonprojectable versions of the theory. We obtain the structure of constraints in each case, and analyze its dependence on the values of the coupling constants involved in the additional constraints. In the nonprojectable theory, the scalar graviton is absent when the coupling constants have certain values, while for other values the scalar graviton appears. The projectable theory is free from the scalar graviton regardless of the values of the coupling constants, even though the structure of the constraints does depend on the value of a coupling constant.