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Energy band structure of the bilayer graphene superlattices withzero-averaged periodic $\delta$-function potentials are studied within thefour-band continuum model. Using the transfer matrix method, studies are mainlyfocused on examining the touching points between adjacent minibands. For thezero-energy touching points the dispersion relation derived shows theDirac-like double-cone shape with the group velocity which is periodic in thepotential strength $P$ with the period of $\pi$ and becomes anisotropic atrelatively large $P$. From the finite-energy touching points we have identifiedthose located at zero wave-number. It was shown that for these finite-energytouching points the dispersion is direction-dependent in the sense that it islinear or parabolic in the direction parallel or perpendicular to thesuperlattice direction, respectively. We have also calculated the density ofstates and the conductivity that demonstrate a manifestation of the touchingpoints of interest.

Touching points in the energy band structure of bilayer graphene superlattices