  The reduction problem of the chiral field equation on symmetric spaces is
studied. It is shown that the symmetric chiral field has infinitely many local
conservation laws. A recursive formula for these conservation laws is derived
and the first associated integral of motion are given explicitly. Furthermore,
the Zakharov-Mikhailov (inverse scattering) transform is used to derive
explicit formulas for the N-soliton solution on arbitrary diagonal background.
The solitons' properties and interactions are analyzed. Such solitons are
naturally related to gravitational solitons of Einstein's field equations, and
this result is used to clarify why the latter do not have fixed amplitude and
velocities (unlike `classical' solitons). Finally, it is proven that the
symmetric chiral field (matrix) equation is equivalent to a single scalar
equation, which in turn, is equivalent to the Sine-Gordon equation.
