  We consider a connected symplectic manifold $M$ acted on properly and in a
Hamiltonian fashion by a connected Lie group $G$. Inspired to the recent paper
\cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of
Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits
is given and we also describe under which condition a Lagrangian orbit is
isolated. If $M$ is a compact K\"ahler manifold we give a necessary and
sufficient condition to an isometric action admits a Lagrangian orbit. Then we
investigate Lagrangian homogeneous submanifolds on the symplectic cut and on
the symplectic reduction. As an application of our results, we give new
examples of Lagrangian homogeneous submanifolds on the blow-up at one point of
the complex projective space and on the weighted projective spaces. Finally,
applying Proposition \ref{slice} that we may call \emph{Lagrangian slice
theorem} for group acting with a fixed point, we give new examples of
Lagrangian homogeneous submanifolds on irreducible Hermitian symmetric spaces
of compact and noncompact type.
