Symmetric submersions of Rn→RmRn→Rm

Submersions ff such that f−1(0)f−1(0) contains a given fiber FF, and that are invariant under a family of vector fields ss leaving FF invariant, are constructed. Examples for which a submersion of this kind cannot exist are also given. In the absence of a geometric theory of submersions ff invariant under ss, most of our treatment is analytic.