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We consider the two planes at zero temperature with isotropic conductivitythat are in relative lateral motion with velocity $v$ and inter-plane distance$a$. Two models of conductivity are taken into account -- the constant andfrequency-dependent Drude models. The normal (perpendicular to planes) Casimirforce is analysed in detail for two systems -- i) two planes with identicalconductivity and ii) one of the planes is a perfect metal. The velocitycorrection to the Casimir energy $\Delta_v\mathcal{E} \sim v^2$ for smallvelocity for all considered cases. In the case of the constant conductivity$\eta$, the energy correction is $ \Delta_v\mathcal{E} \sim \frac{\eta}{a^3}\left(\frac{v}{\eta}\right)^2$for $v\ll \eta \ll 1$.

The normal Casimir force for lateral moving planes with isotropic conductivities