Baby-Skyrmions dressed by fermions, an analytic sector

We find an analytic solution of the backreacted coupled fermion-baby-Skyrmion system valid at all values of the coupling parameter. The solution, built on a finite cylinder, is generally given in terms of the Heun functions and satisfies the physical requirements of finite energy. For a special value of the coupling parameter, the solution becomes a periodic crystal of baby-Skyrmions and fermions defined on the plane $mathbb{R}^2$. These solutions are trivially extended to multi-solitonic branches of higher Baryon number.