We derive the Euler-Lagrange equations for minimizers of causal variational
principles in the non-compact setting with constraints, possibly prescribing
symmetries. Considering first variations, we show that the minimizing measure is
supported on the intersection of a hyperplane with a level set of a function which
is homogeneous of degree two. Moreover, we perform second variations to obtain
that the compact operator representing the quadratic part of the action is positive
semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange
multiplier method to variational principles on convex sets.