We present Lie symmetry analysis to explore solitary wave solutions, two-soliton type solutions and three-soliton type solutions in variable-coefficient nonlinear physical phenomena. An example is a (2+1)-dimensional variable-coefficient Bogoyavlensky-Konopelchenko (VCBK) equation. We compute the Lie algebra of infinitesimals of its symmetry vector fields and an optimal system of one-dimensional sub-Lie algebras of the resulting symmetries. Two stages of Lie symmetry reductions will be built to reduce the VCBK equation to nonlinear ordinary differential equations (ODEs) and new analytical solutions to those ODEs will be found by using the integration method. Some of such resulting solutions to the VCBK equation and their dynamics will be illustrated through three-dimensional plots.