  In this work, we aim mainly to study the existence of different classes of
standing waves for a class of nonlinear Schr\"odinger system with quadratic
interaction, combined with a harmonic interaction potential or partial harmonic
potential. We establish the existence of ground-state normalized solutions for
this system, serving as local minimizers for associated functionals. Novel
approaches are employed to address challenges posed by the potential term,
including profile decomposition and concentration-compactness principles. The
absence of global energy minimizers in critical and supercritical cases
necessitates focusing on local energy minimizers. Notably, positive results are
observed in cases of partial confinement due to the spectral properties of
associated linear operators. Additionally, we demonstrate the existence of a
second normalized solution using Mountain-pass geometry, overcoming challenges
posed by nonlinear terms. Additionally, the asymptotic behavior of local
minimizers is explored, revealing connections with unique eigenvectors of the
linear operators. The analysis showcases the dispersive properties of
Schr\"odinger operators and derives Strichartz estimates for local
well-posedness in the energy space. Global and blow-up solutions are derived in
time under certain conditions, contributing new insights into the dynamics of
the system.
