  We construct self-dual sectors for scalar field theories on a
$(2N+2)$-dimensional Minkowski space-time with target space being the
$2N+1$-dimensional sphere $S^{2N+1}$. The construction of such self-dual
sectors is made possible by the introduction of an extra functional on the
action that renders the static energy and the self-duality equations
conformally invariant on the $(2N+1)$-dimensional spatial submanifold. The
conformal and target space symmetries are used to build an ansatz that leads to
an infinite number of exact self-dual solutions with arbitrary values of the
topological charge. The five dimensional case is discussed in detail where it
is shown that two types of theories admit self dual sectors. Our work
generalizes the known results in the three-dimensional case that leads to an
infinite set of self-dual Skyrmion solutions.
