We study existence and nonexistence of positive, spherically symmetric solutions of diagonal quasilinear elliptic systems involving equations with\break $p$-Laplacians, and with strong dependence on the gradient on the right-hand side. The existence proof is constructive, with solutions possessing explicit integral representation. Also, we obtain critical exponents of the gradient. We introduce the notion of cyclic elliptic systems in order to study nonsolvability of general elliptic systems. The elliptic system is studied by relating it to the corresponding system of singular ordinary integro-differential equations of the first order.