  A $2n$-dimensional Lax integrable system is proposed by a set of specific
spectral problems. It contains Takasaki equations, the self-dual Yang-Mills
equations and its integrable hierarchy as examples. An explicit formulation of
Darboux transformations is established for this Lax integrable system. The
Vandermonde and generalized Cauchy determinant formulas lead to a description
for deriving explicit solutions and thus some rational and analytic solutions
are obtained.
