A variable coefficient nonlinear Schr\"{o}dinger equation with a four-dimensional symmetry group and blow-up of its solutions

A canonical variable coefficient nonlinear Schr"{o}dinger equation with a four dimensional symmetry group containing $\text{SL}(2,\mathbb{R})$ group as a subgroup is considered. This typical invariance is then used to transform by a symmetry transformation a known solution that can be derived by truncating its Painlev'e expansion and study blow-ups of these solutions in the $L_p$-norm for $p>2$, $L_{\mathrm{\infty }}$-norm and in the sense of distributions.