harmonicity and submanifold maps

The aim of this paper is fourfold. firstly, we introduce and study the fultra-harmonic maps. secondly, we recall the geometric dynamics generated by a firstorder normal pde system and we give original results regarding the geometric dynamicsgenerated by other first order pde systems. thirdly, we determine the gauss pdesand the fundamental forms associated to integral manifolds of first order pde systems.fourthly, we change the gauss pdes into a geometric dynamics on the jet bundle oforder one, showing that there exist an infinity of riemannian metrics such that the liftof a submanifold map into the first order jet bundle to be an ultra-potential map.