First-order formalism for Alice string

We apply the {\it first-order formalism} method to obtaining BPS equations for Alice string. This is done by generalizing the well-known first-order formalism to the case of non-Abelian strings. We do not assume any specific gauge group nor the shape of the kinetic term function, but require only that the fields are axially-symmetric and static. With this formalism we reproduce the BPS equations of $SU(2)\times U(1)$ Alice strings \cite{Chatterjee:2017jsi}, and present their corresponding numerical solutions.