New maximal dimension of invariant subspaces to coupled systems with two-component equations

•A new extension of the theorem “on maximal dimension” to the vector operators under certain conditions is given.•Several concrete examples are presented to illustrate this result.•The complete classification of the Torrisi’s system is obtained.

In this paper, new maximal dimension of invariant subspaces to coupled systems with two-component equations is estimated under certain conditions. It is shown that if the really coupled operator F=(F1,F2)F=(F1,F2) with orders {k1,k2}(k1⩾k2){k1,k2}(k1⩾k2) preserves the invariant subspace Wn11×Wn22(0