  We define quantum determinants in Quantum Matrix Algebras, related to couples
of compatible braidings following the scheme from [G]. We establish relations
between these determinants and the so-called column-(row-)determinants, often
used in the theory of integrable systems. Also, we generalize the quantum
integrable spin systems from [CFRS] by using generalized Yangians, related to
couples of compatible braidings. We demonstrate that such quantum integrable
spin systems are not uniquely determined by the "quantum coordinate ring" of
the basic space V. For instance, the "quantum plane" xy=qyx gives rise to two
different integrable systems: rational and trigonometric ones.
