  Metric independent $\sigma$ models are constructed. These are field theories
which generalise the membrane idea to situations where the target space has
fewer dimensions than the base manifold. Instead of reparametrisation
invariance of the independent variables, one has invariance of solutions of the
field equations under arbitrary functional redefinitions of the field
quantities. Among the many interesting properties of these new models is the
existence of a hierarchical structure which is illustrated by the following
result. Given an arbitrary Lagrangian, dependent only upon first derivatives of
the field, and homogeneous of weight one, an iterative procedure for
calculating a sequence of equations of motion is discovered, which ends with a
universal, possibly integrable equation, which is independent of the starting
Lagrangian. A generalisation to more than one field is given.
