  Using a new method and additional (conditional and partial) equivalence
transformations, we performed group classification in a class of variable
coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the
general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of
such equations with the density $f$ localized in space, which have large
invariance algebra. Exact solutions of these equations are constructed. We also
consider the problem of investigation of the possible local trasformations for
an arbitrary pair of equations from the class under consideration, i.e. of
describing all the possible partial equivalence transformations in this class.
