Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear pdes

Given a G s -involutive structure, ( M , V ) , a Gevrey submanifold X ⊂ M which is maximally real and a Gevrey function u 0 on X we construct a Gevrey function u which extends u 0 and is a Gevrey approximate solution for V . We then use our construction to study Gevrey micro-local regularity of solutions, u ∈ C 2 ( R N ) , of a system of nonlinear pdes of the form F j ( x , u , u x ) = 0 , j = 1 , … , n , where F j ( x , ζ 0 , ζ ) are Gevrey functions of order s > 1 and holomorphic in ( ζ 0 , ζ ) ∈ C × C N . The functions F j satisfy an involutive condition and d ζ F 1 ∧ ⋯ ∧ d ζ F n ≠ 0 .