  We study two-body interactions of magnetic skyrmions on the plane and apply
them to a (mostly) analytic description of a skyrmion lattice. This is done in
the context of the solvable line, a particular choice of a potential for
magnetic anisotropy and Zeeman terms, where analytic expressions for skyrmions
are available. The energy of these analytic single skyrmion solutions is found
to become negative below a critical point, where the ferromagnetic state is no
longer the lowest energy state. This critical value is determined exactly
without the ambiguities of numerical simulations. Along the solvable line the
interaction energy for a pair of skyrmions is repulsive with power law fall off
in contrast to the exponential decay of a purely Zeeman potential term. Using
the interaction energy expressions we construct an inhomogeneous skyrmion
lattice state, which is a candidate ground states for the model in particular
parameter regions. Finally we estimate the transition between the skyrmion
lattice and an inhomogeneous spiral state.
