  Some new examples of K-monotone couples of the type (X, X(w)), where X is a
symmetric space on [0, 1] and w is a weight on [0, 1], are presented. Based on
the property of the w-decomposability of a symmetric space we show that, if a
weight w changes sufficiently fast, all symmetric spaces X with non-trivial
Boyd indices such that the Banach couple (X, X(w)) is K-monotone belong to the
class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental
function of X is t^{1/p} for some p \in [1, \infty], then X = L_p. At the same
time a Banach couple (X, X(w)) may be K-monotone for some non-trivial w in the
case when X is not ultrasymmetric. In each of the cases where X is a Lorentz,
Marcinkiewicz or Orlicz space we have found conditions which guarantee that (X,
X(w)) is K-monotone.
