  We systematically studied the spin-lattice couplings in the CrF$_3$
monolayer. Our study reveals that the spin exchange constants between the
nearest neighbors are notably affected by these couplings. Specifically, the
couplings arise predominantly from three distinct phonon modes, namely the
covariant, rotation, and stretch of the Cr-F-Cr-F rhombus. By integrating out
the phonon degrees of freedom, we derived an effective spin Hamiltonian
featuring four-spin product terms, which yields a remarkably intricate magnetic
phase diagram. Significantly, numerous plateau states characterized by
fractional magnetizations, including 1/2, 1/3, 2/3, 1/4, 1/5, 5/8, 1/9, and
2/9, emerge in the vicinity of the phase transition boundary separating
ferromagnetic and antiferromagnetic states. These findings show the profound
influence of spin-lattice couplings on magnetic properties near the magnetic
phase boundaries, and the predicted plateau states are expected to be
observable in future experiments.
