  We study the existence of $L^2$ normalized solutions for nonlinear
Schr\"odinger equations and systems. Under new Palais-Smale type conditions we
develop new deformation arguments for the constraint functional on $S_m=\{ u;
\, \int_{\mathbf{R}^N} | u |^2=m\}$ or $S_{m_1} \times S_{m_2}$. As
applications, we give other proofs to the results of [\cite[J:20],
\cite[BdV:6], \cite[BS1:7]]. As to the results of [\cite[J:20], \cite[BdV:6]],
our deformation result enables us to apply the genus theory directly to the
corresponding functional to obtain infinitely many solutions. As to the result
[\cite[BS1:7]], via our deformation result we can show the existence of vector
solution without using constraint related to the Pohozaev identity.
