In this paper we prove the existence of a solution in \( L_{{\text{loc}}}^\infty \left( \Omega \right) \) to the Euler-Lagrange equation for the variational problem $$ \mathop {\inf }\limits_{\bar u + w_0^{1,\infty } \left( \Omega \right)} {\mathbf{ }}\int {_\Omega } \left( {1_D \left( {\nabla u} \right) + g\left( u \right)} \right)dx, $$ (1) with D convex closed subset of ℝn with non empty interior. By means of a disintegration theorem, we next show that the Euler-Lagrange equation can be reduced to an ODE along characteristics, and we deduce that the solution to Euler-Lagrange is different from 0 a.e. and satisfies a uniqueness property. Using these results, we prove a conjecture on the existence of variations on vector fields [3].