  An orbit-like foliation is a singular foliation on a complete Riemannian
manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian
foliation) and (transversely) infinitesimally homogenous. This class of
singular foliation contains not only the classe of partion of the space into
orbits of isometric actions, but also infinite many non homogenous examples and
in particular the partition of $M$ into orbits of a proper groupoid.
  In this paper we prove a version of Kondrakov Embedding Theorem and an
analogous Principle of Symmetric Criticality of Palais for basic funcions of
orbit-like foliations. As proof of concepts, we study not only the
corresponding Yamabe problem in the setting, but also to the case of fiber
bundles with homogeneous fibers, seeking for the existence of metrics with
constant scalar curvature that respect the respective Riemannian Foliation
decomposition. An application to the existence of positive constant scalar
curvature on exotic spheres is presented. In an upcoming version we shall
extend the results to the corresponding Kazdan--Warner problem.

Differential Geometry;Analysis of PDEs