if formed, is $ca.$ four times less mobile than $x$ and $y$ bipolarons. The bipolaron bandwidth is *of the same order* as the polaron one, which is a specific feature of the inter-site bipolaron. For a large part of the Brillouin zone near $%
(0,\pi )$ for ‘$x$’ and $(\pi ,0)$ for ‘$y$’ bipolarons, one can adopt the effective mass approximation $$E^{x,y}({\bf K})={\frac{K_{x}^{2}}{{2m_{x,y}^{\ast \ast }}}}+{\frac{%
K_{y}^{2}}{{2m_{y,x}^{\ast \ast }}}}$$ with $K_{x,y}$ taken relative to the band bottom positions and $m_{x}^{\ast \ast }=1/t$, $m_{y}^{\ast \ast }=4m_{x}^{\ast \ast }$.

$X$ and $y$ bipolarons bose-condense at the boundaries of the center-of-mass Brillouin zone with $K=(\pm\pi,0)$ and $K=(0,\pm\pi)$, respectively, which explains the d-wave symmetry and the checkerboard modulations of the order parameter in cuprates [@aledwave].

In-plane bipolarons
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Now let us consider in-plane bipolarons in a two-dimensional lattice of ideal octahedra that can be regarded as a simplified model of the copper-oxygen perovskite layer, Fig.3 [@alekor2]. The lattice period is $a=1$ and the distance between the apical sites and the central plane is $h=a/2=0.5$. For mathematical transparency we assume that all in-plane atoms, both copper and oxygen, are static but apex oxygens are independent three-dimensional isotropic harmonic oscillators.

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Due to poor screening, the hole-apex 