# Number of Weyl points due to symmetries

In Weyl semimetals time reversal symmetry (TRS) or centrosymmetry has to be broken (CS). It is stated, there are at least four Weyl points in TRS systems. I tried understanding this looking at the semiclassical equation of motion

$$\dot{\vec{r}} = \vec{v_k} + \dot{\vec{k}} \times \vec{\Omega_k}$$

Under time reversion $$t \rightarrow -t$$ , $$\vec{v_k} \rightarrow -\vec{v}_{-k}$$ and $$\dot{\vec{k}} \rightarrow \dot{\vec{k}}$$. For TRS, it follows that the Berry curvature should transform such that $$\vec{\Omega_k} \rightarrow -\vec{\Omega}_{-k}$$. In that case, the equation above is satisfied.

However, I don't see the argument, why there have to be at least four Weyl points in TRS systems, since the Berry curvature transforms by switching sign and thus describes a Weyl point with opposite chirality at $$-\vec{k}$$.

Is there TRS if the equation above is satisfied after applying the transformation? Or does the TRS actually mean, that the Berry curvature should not transform like mentioned above. I guess I amm mixing up the transformation itself and the corresponding symmetry.

In some papers I read that for TRS systems the Berry curvature holds

$$\vec{\Omega_k} = - \vec{\Omega_{-k}}$$

That is, there should be a Weyl point with opposite chirality and no-go theorem is not violated. Why there has to be at least four Weyl points? On the other hand I found in one paper for TRS systems

$$\vec{\Omega_k} = \vec{\Omega_{-k}}$$