{#int}
-----------------------------

Our main interest is the C-S model with weight $$\begin{aligned}
\label{psi}
\psi(s)=
\left\{
\begin{array}{ccc}
s^{-\alpha}&{\rm for}& s>0,\\
0&{\rm for}& s=0,
\end{array}
\ \ \ \ \ \ \alpha\in(0,d).\right.\end{aligned}$$ With the loss of Lipschitz continuity of $\psi$ the problem of existence and uniqueness for the discrete C-S model becomes more difficult. To our best knowledge there are no results in this direction, even though the are results on asymptotics in such case, see e.g. [@haliu]. When dealing with the C-S model with bounded weight $\psi_{cs}$ one makes use of Lipschitz continuity of $\psi_{cs}$ as well as the structure of the model itself. As an example we will now present a simple aplication of the properties of the structure of our model. Namely we will prove that the avarage velocity of the particles $$\begin{aligned}
\bar{v}(t):=\frac{1}{N}\sum_{i=1}^Nv_i(t)\end{aligned}$$ is constant in time. Assuming that $x=(x_1,...,x_N)$ and $v=(v_1,...,v_N)$ is a sufficiently smooth solution of (\[cs1\]) and (\[cs2\]), we calculate the derivative of $\bar{v}$ to get $$\begin{aligned}
\frac{d}{dt}\sum_{i=1}^Nv_i=\frac{1}{N}\sum_{i,k=1}^N(v_k-v_i)\psi_n(|x_i-x_k|)=\\
\frac{1}{2N}\sum_{i,k=1}^N(v_k-v_i)\psi_n(|x_i-x_k|)+ \frac{1}{2N}\sum_{i,k=1}^N(v_i-v_k)\psi_n(|x_i-x_k|)=0,\end{aligned}$$ where the latter summant in the second line is obtained by substituting $i$ and $k$. Clearly each such structure based property of the C-S model will remain true regardless of the communication weight $\psi$ as 