  We study a recently proposed modification of the Skyrme model that possesses
an exact self-dual sector leading to an infinity of exact Skyrmion solutions
with arbitrary topological (baryon) charge. The self-dual sector is made
possible by the introduction, in addition to the usual three SU(2) Skyrme
fields, of six scalar fields assembled in a symmetric and invertible three
dimensional matrix h. The action presents quadratic and quartic terms in
derivatives of the Skyrme fields, but instead of the group indices being
contracted by the SU(2) Killing form, they are contracted with the h-matrix in
the quadratic term, and by its inverse on the quartic term. Due to these extra
fields the static version of the model, as well as its self-duality equations,
are conformally invariant on the three dimensional space R^3. We show that the
static and self-dual sectors of such a theory are equivalent, and so the only
non-self-dual solution must be time dependent. We also show that for any
configuration of the Skyrme SU(2) fields, the h-fields adjust themselves to
satisfy the self-duality equations, and so the theory has plenty of non-trivial
topological solutions. We present explicit exact solutions using a holomorphic
rational ansatz, as well as a toroidal ansatz based on the conformal symmetry.
We point to possible extensions of the model that break the conformal symmetry
as well as the self-dual sector, and that can perhaps lead to interesting
physical applications.
