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We study two dimensional soliton solutions in the $CP^2$ nonlinear$\sigma$-model with a Dzyaloshinskii-Moriya type interaction. First, we derivesuch a model as a continuous limit of the $SU(3)$ tilted ferromagneticHeisenberg model on a square lattice. Then, introducing an additional potentialterm to the derived Hamiltonian, we obtain exact soliton solutions forparticular sets of parameters of the model. The vacuum of the exact solutioncan be interpreted as a spin nematic state. For a wider range of couplingconstants, we construct numerical solutions, which possess the same type ofasymptotic decay as the exact analytical solution, both decaying into a spinnematic state.

Isolated Skyrmions in the $CP^2$ nonlinear $\sigma$-model with a Dzyaloshinskii-Moriya type interaction