Oscillation of solutions of a system of differential equations

The system
$$ u'_1=a_1(t)|u_2|^{\lambda_1}\operatorname{sign}u_2,\qquad u'_2=-a_2(t)|u_1|^{\lambda_2}\operatorname{sign}u_1,\eqno(1) $$
is considered, where the functions $a_i:[0,+\infty)\to\mathbf R$ $(i=1,2)$ are locally summable, $\lambda_i>0$ $(i=1,2)$ and $\lambda_1\cdot\lambda_2=1$. Sufficient conditions are obtained for all solutions of system (1) to be oscillating. Furthermore, functions $a_i(t)$ $(i=1,2)$ are, generally speaking, not assumed to be nonnegative.