Local solutions of systems of semi-linear second order partial differential equations

We show that one can construct a germ of a local solution of systems of semi-linear second order partial differential equations (PDEs) from any given n -jet that satisfies the system up to the ( n − 2 ) -jet level. We treat in more details the case of harmonic map-germs to the unit sphere. Our results allow, for instance, to determine which local singularities a harmonic map to the sphere can have.