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We made heat-capacity measurements of two dimensional (2D) $^3$He adsorbed ongraphite preplated with monolayer $^4$He in a wide temperature range (0.1 $\leqT \leq$ 80 mK) at densities higher than that for the 4/7 phase (= 6.8nm$^{-2}$). In the density range of 6.8 $\leq \rho \leq$ 8.1 nm$^{-2}$, the 4/7phase is stable against additional $^3$He atoms up to 20% and they are promotedinto the third layer. We found evidence that such promoted atoms form aself-bound 2D Fermi liquid with an approximate density of 1 nm$^{-2}$ from themeasured density dependence of the $\gamma$-coefficient of heat capacity. Wealso show evidence for the first-order transition between the commensurate 4/7phase and the ferromagnetic incommensurate phase in the second layer in thedensity range of 8.1 $\leq \rho \leq$ 9.5 nm$^{-2}$.

Evidence for a Self-Bound Liquid State and the Commensurate-Incommensurate Coexistence in 2D $^3$He on Graphite