  We suggest to use for $XY_2$ molecules some results previously established in
a series of articles for vibrational modes and electronic states with an $E$
symmetry type. We first summarize the formalism for the standard $u(2)\supset
su(2)\supset so(2)$ chain which, for its most part, can be kept for the study
of both stretching and bending modes of $XY_2$ molecules. Next the also
standard chain $u(3)\supset u(2) \supset su(2) \supset so(2)$ which is
necessary, within the considered approach, is introduced for the stretching
modes. All operators acting within the irreducible representation
(\textit{irrep}) $[N00]\equiv [N\dot{0}]$ of $u(3)$ are built and their matrix
elements computed within the standard basis. All stretch-bend interaction
operators taking into account the polyad structure associated with a resonance
$\omega_1\approx \omega_3 \approx 2 \omega_2$ are obtained. As an illustration,
an application to the $D_2S$ molecular system is considered, especially the
symmetrization in $C_{2v}$. It is shown that our unitary formalism allows to
reproduce in an extremely satisfactory way all the experimental data up to the
dissociation limit.
