  Miura surfaces are the solutions of a constrained nonlinear elliptic system
of equations. This system is derived by homogenization from the Miura fold,
which is a type of origami fold with multiple applications in engineering. A
previous inquiry, gave suboptimal conditions for existence of solutions and
proposed an $H^2$-conformal finite element method to approximate them. In this
paper, the existence of Miura surfaces is studied using a mixed formulation. It
is also proved that the constraints propagate from the boundary to the interior
of the domain for well-chosen boundary conditions. Then, a numerical method
based on a least-squares formulation, Taylor--Hood finite elements and a Newton
method is introduced to approximate Miura surfaces. The numerical method is
proved to converge and numerical tests are performed to demonstrate its
robustness.
