The interplay between two Euler–Lagrange operators relating to the nonlinear elliptic system $$\Sigma [(u, {\mathscr {P}}), \varOmega ]$$

We establish the existence of multiple whirling solutions to a class of nonlinear elliptic systems in variational form subject to pointwise gradient constraint and pure Dirichlet type boundary conditions. A reduced system for certain $$\mathbf{SO}(n)$$ SO ( n ) -valued matrix fields, a description of its solutions via Lie exponentials, a structure theorem for multi-dimensional curl free vector fields and a remarkable explicit relation between two Euler–Lagrange operators of constrained and unconstrained types are the underlying tools and ideas in proving the main result.