Exact solutions of the elliptic sine–cosine equation ∂2ψ/∂x2+∂2ψ/∂y2=sin(ψ+g) are derived in two space dimensions with the aid of a new Bäcklund transformation and by exploiting the properties of the harmonic function g (x,y). Two generating formulas are developed which allow us to generate without additional quadratures an infinite number of real solutions α and infinitely many imaginary solutions iβ. Some α solutions behave like solitons and can be labeled by a topological quantum number. Which solutions are solitonlike and which are not depends decisively on the analytic structure of g and its domain of harmonicity in the ℛ2 plane.