In the following paper we are going to study a special kind of solitons, called vortices. We will define the topological number for them, called the vortex number. Switching to complex variables and using a different approach from the one used by Jaffe and Taubes in the famous monograph "Vortices and monopoles", 1980, in particular, using the theorems known in pseudoanalytic function theory, we construct the vortex solutions of the field equations for our problem and write out the vortex numbers for them that will be characterized by the zeroes of each possible solution.