  We explain how to apply techniques from integrable systems to construct
$2k$-soliton homoclinic wave maps from the periodic Minkowski space $S^1\times
R^1$ to a compact Lie group, and more generally to a compact symmetric space.
We give a correspondence between solutions of the -1 flow equation associated
to a compact Lie group $G$ and wave maps into $G$. We use B\"acklund
transformations to construct explicit $2k$-soliton breather solutions for the
-1 flow equation and show that the corresponding wave maps are periodic and
homoclinic. The compact symmetric space $G/K$ can be embedded as a totally
geodesic submanifold of $G$ via the Cartan embedding. We prescribe the
constraint condition for the -1 flow equation associated to $G$ which insures
that the corresponding wave map into $G$ actually lies in $G/K$. For example,
when $G/K=SU(2)/SO(2)=S^2$, the constrained -1-flow equation associated to
SU(2) has the sine-Gordon equation (SGE) as a subequation and classical
breather solutions of the SGE are 2-soliton breathers. Thus our result
generalizes the result of Shatah and Strauss that a classical breather solution
of the SGE gives rise to a periodic homoclinic wave map to $S^2$. When the
group $G$ is non-compact, the bi-invariant metric on $G$ is pseudo-Riemannian
and B\"acklund transformations of a smooth solution often are singular. We use
B\"acklund transformations to show that there exist smooth initial data with
constant boundary conditions and finite energy such that the Cauchy problem for
wave maps from $R^{1,1}$ to the pseudo-Riemannian manifold $SL(2,R)$ develops
singularities in finite time.
