satisfied.

5.  Points $P_{7,8}$: Same as $P_6$, they would not exist if $f(\lambda_a)\neq 0$.

6.  Point $P_{9,10}$: $f(\lambda_a)=0$ should hold. The fixed values and the eigenvalues of these two points depend on $f_1(\lambda_a)=
    \left.\frac{df(\lambda)}{d\lambda}\right|_{\lambda=\lambda_a}$.

Thus, only points $P_{1,2}$ are independent of $f(\lambda)$.

Comparing the fixed points in LQC and the ones in classical cosmology (see the Table **I** of [@Fang-CQG]), we can see that, even though the values of the coordinates $(x_{c},y_{c},\lambda_{c})$ are the same, the stability properties are very different. This is reasonable, because the quantum modification is considered, and the autonomous system in the LQC scenario is very different from the one in the classical scenario, e.g., the autonomous system is four dimensional in LQC but three dimensional in the classical scenario. Notice that all of the fixed points lie in the classical regions, and therefore the coordinates of fixed points remain the same from classical to LQC, which we also pointed out in an earlier paper [@Xiao-dynamical].

Now we focus on the late time attractors: point $P_{3}$ under the conditions of $\gamma=1, f_1(0)\geq 0$ and $\gamma=4/3, f_1(0)=0$, and point $P_{9}$ under the conditions of $\lambda_a^2<6,
f_1(\lambda_a)>\lambda_a,\lambda_a<3\gamma$. Obviously, these points are scalar-field dominated, since ${\rho_\gamma}=H^2(1/(1-z_c)-x_c^2-y_c^2)=0$. For point $P_{3}$, 