Description of solutions with the uniton number $3$ in the case of one eigenvalue: Counterexample to the dimension conjecture

We explicitly describe solutions of the noncommutative unitary $U(1)$ sigma model that represent finite-dimensional perturbations of the identity operator and have only one eigenvalue and the minimum uniton number $3$. We also show that the solution set $M(e,r,u)$ of energy $e$ and canonical rank $r$ with the minimum uniton number $u=3$ has a complex dimension greater than $r$ for $e=4n-1$ and $r=n+1$, where $n\ge3$. This disproves the dimension conjecture that holds in the case $u\in\{1,2\}$.