Uniqueness of the ODE solution

Question

There is a system $$\frac{dx}{dt} = y^{3}+ ln(t+1) , \ x\frac{dy}{dt}=(y-t)^{\frac{1}{3}}$$ It has the only solution by the Cauchy – Kovalevskaya theorem under the following initial conditions: $$t_{0}>-1, \ x_{0}\neq 0, \ y_{0}\neq t_{0}$$ Will the system have the only solution if $$y_{0}=t_{0}$$ (other initial conditions will not change) ?