  Coupled nonlinear integrable systems are generated from usual zero curvature
equation. The relevant Maurer-Cartan forms are constructed by combining
suitably chosen matrices (nilpotent, Hadamard, idempotent and k-idempotent) and
Lie algebraic elements via Kronecker product. In each case a closure type
property among the matrices chosen is found to be playing a key role to produce
both the coupling and nonlinearity present in the system of equations obtained.
The method is highly flexible and can be used to construct general systems
containing 'p' number of equations. It is also shown that these new equations
can be written in the Hamiltonian form (with a preassigned symplectic operator)
with the trace identity introduced by Tu. Since the Lax operator is known one
can obtain the hereditary operators signifying the complete integrability.
Various properties of Kronecker product are found to be useful in our
construction.
