General solutions of the supersymmetric $\mathbb{C}P^2$ sigma model and its generalisation to $\mathbb{C}P^{N-1}$

A new approach for the construction of finite action solutions of the supersymmetric $\mathbb{C}P^{N-1}$ sigma model is presented. We show that this approach produces more non-holomorphic solutions than those obtained in previous approaches. We study the $\mathbb{C}P^2$ model in detail and present its solutions in an explicit form. We also show how to generalise this construction to $N>3$.