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In this paper we consider an energy functional depending on the norm of the gradient and seek to extremise it over an admissible class of Sobolev maps defined on an annulus and taking values on the unit sphere whilst satisfying suitable boundary conditions. We establish the existence of an infinite family of solutions with certain symmetries to the associated nonlinear Euler-Lagrange system in even dimensions and discuss the stability of such extremisers by way of examining the positivity of the second variation of the energy at these solutions.

A class of extremising sphere-valued maps with inherent maximal tori symmetries in SO ( n ) $\mathbf{SO}(n)$