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

Given a GsGs-involutive structure  , (M,V)(M,V), a Gevrey submanifold X⊂MX⊂M which is maximally real and a Gevrey function u0u0 on X we construct a Gevrey function u   which extends u0u0 and is a Gevrey approximate solution for VV. We then use our construction to study Gevrey micro-local regularity of solutions, u∈C2(RN)u∈C2(RN), of a system of nonlinear pdes of the formFj(x,u,ux)=0,j=1,…,n, where Fj(x,ζ0,ζ)Fj(x,ζ0,ζ) are Gevrey functions of order s>1s>1 and holomorphic in (ζ0,ζ)∈C×CN(ζ0,ζ)∈C×CN. The functions FjFj satisfy an involutive condition and dζF1∧⋯∧dζFn≠0dζF1∧⋯∧dζFn≠0.