  We study the symmetry group properties of the variable coefficient
Davey-Stewartson (vcDS) system. The Lie point symmetry algebra with a
Kac-Moody-Virasoro (KMV) structure is shown to be isomorphic to that of the
usual (constant coefficient) DS system if and only if the coefficients satisfy
some conditions. These conditions turn out to coincide with those for the vcDS
system to be transformable to the DS system by a point transformation.
  The equivalence group of the vcDS system is applied to pick out the
integrable subsystems from a class of non-integrable ones. Additionally, the
full symmetry group of the DS system is derived explicitly without
exponentiating its symmetry algebra.
  Lump solutions (rationally localized in all directions in $\mathbb{R}^2$)
introduced by Ozawa for the DS system is shown to hold even for the vcDS system
precisely when the system belongs to the integrable class, i.e. equivalent to
the DS system. These solutions can be used for establishing exact blow-up
solutions in finite time in the space $L^2(\mathbb{R}^2)$ in the focusing case.
