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In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form $$y_i^{\text{'}}\left(t\right)-p_i\left(t\right)y_{i+1}\left(t\right)=0,i=1,2,\cdots ,n-2,y_{n-1}^{\text{'}}\left(t\right)-p_{n-1}\left(t\right)|y_n\left(h_n\left(t\right)\right){|}^{\alpha }\mathrm{s}gn\left[y_n\left(h_n\left(t\right)\right)\right]=0,y_n^{\text{'}}\left(t\right)\mathrm{s}gn\left[y_1\left(h_1\left(t\right)\right)\right]+p_n\left(t\right){\left|y_1\left(h_1\left(t\right)\right)\right|}^{\beta }\le 0,$$ where $n\ge 3$ is odd, $\alpha >0$, $\beta >0$.