A counterexample to a conjecture on the dimension of the solution spaces of the non-commutative sigma-model

The conjecture on the dimension of the solution spaces of the non-commutative $U(1)$ sigma-model asserts that the set of all solutions of canonical rank $r$ has complex dimension $r$. This conjecture holds true for solutions of uniton number 1 and 2. We construct a series of examples of complex $k$-dimensional families of solutions with uniton number 3 and canonical rank $r$ such that $k>r$ and the difference $k-r$ may be arbitrarily large.