The paper deals with the differential operator \(L\) generated in the space \(L_2^m(\mathbb{R})\) of vector-valued functions by the formally self-adjoint differential expression \[(-i)^{2\nu}y^{(2\nu)}(x)+\displaystyle\sum_{k=2}^{2\nu}P_k(x)y^{(2\nu-k)}(x),\] where \(\nu>1\) and \(P_k(x)\) is a \(m\times m\) matrix with summable entries \(p_{k,i,j}\) which satisfy the periodicity conditions \(p_{k,i,j}(x+1)=p_{k,i,j}(x)\) for all \(i=1,2,\ldots,m\) and \(j=1,2,\ldots,m\), for \(k=2,3,\ldots,2\nu\). The author investigates the band functions, Bloch functions and the spectrum of operator \(L\).