Let \(n\), \(N\in\mathbb{N}\), \(n\geq 2\), \(N\geq 1\), \(T> 0\) and \(\mathbb{R}^+= (0,\infty)\). The authors consider the \(n\)th-order differential system in \(\mathbb{R}^N\) \[ x^{(n)}+ A_{n-1} x^{(n- 1)}+\cdots+ A_1x'+ A_0f(t, x)= e(t)\tag{1} \] together with the periodic boundary conditions \[ x(0)= x(T),\quad x'(0)= x'(T),\dots, x^{(n-1)}(0)= x^{(n- 1)}(T).\tag{2} \] Here, \(A_0,A_1,\dots, A_{n-1}\) are real constant \(N\times N\)-matrices, with \(A_0\) nonsingular, \(f: [0,T]\times (\mathbb{R}^+)^N\to \mathbb{R}^N\) satisfies the \(L_1\)-Carathéodory conditions, \(f\) may be singular at the value \(0\) of its phase variable and \(e\in L_1(0,T; \mathbb{R}^N)\) is such that \(\int^T_0 e(t)\,dt= 0\). By a solution of problem (1), (2) we mean a function \(x\in W^{n,1}(0,T; \mathbb{R}^N)\), with \(\min x_i> 0\) for each \(i\in\{1,\dots, N\}\), which satisfies (1) a.e. on \([0, T]\) and the periodicity conditions (2). Under the assumptions that all solutions of the equation \[ x^{(n)}+ A_{n-1} x^{(n-1)}+\cdots+ A_1 x'= 0 \] satisfying (2) are constant and \(f\) satisfies the conditions of Landesman and Lazer type and further growth conditions, the existence of a solution of problem (1), (2) is proved. The proof is based on the Mawhin coincidence degree theory.