  The motion of test membranes on which the group $G$ of isometries of a
spacetime $M$ acts has been considered in general settings. It has been shown
that the configuration of Nambu-Goto membranes is described by the Nambu-Goto
membranes in the quotient manifold $M/G$ with an appropriate projected metric
if (i) $G$ is Abelian, (ii) $G$ is semisimple and compact, or (iii) the
orthogonal distribution of the orbit of $G$ is integrable, but in general not.
  It has also been shown that a similar result holds when the membranes couple
with scalar maps or differential form fields.
