Formulations of axisymmetric equilibrium problems are presented in orthogonal flux coordinates with boundary conditions that do not assume fixation of the boundary shape for both levitron-type traps with a plasma-embedded conductor and conductor-free configurations, including a tokamak. Illustrative examples of numerical solutions of these problems are presented. For traps with a conductor, it is demonstrated how the geometry of the equilibrium with an isodynamic magnetic surface can be found using flux coordinates.