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HE 0107$-$5240 is a hyper metal-poor star with $\rm [Fe/H]=-5.39$. Weperformed high-res observations with the ESPRESSO spectrograph at the VLT toconstrain the kinematical properties of the binary system HE 0107$-$5240 and toprobe the binarity of the sample of 8 most metal-poor stars with $\rm[Fe/H]<-4.5$. Radial velocities are obtained by using cross-correlation in theinterval 4200$-$4315A, which contains the strong CH band, against a template inan iterative process. A Bayesian method is applied to calculate the orbit byusing the ESPRESSO measurements and others from the literature. A chemicalanalysis has also been performed in HE0107$-$5240 by means of spectralsynthesis. Observations of HE 0107$-$5240 spanning more than 3 years show amonotonic decreasing trend in radial velocity at a rate of approximately by 0.5ms$^{-1}$d$^{-1}$. The period is constrained at $P_{\rm orb} =13009_{-1370}^{+1496}$d. New more stringent upper-limits have been found forseveral elements: a)[Sr/Fe] and [Ba/Fe] are lower than $-0.76$ and $+0.2$respectively, confirming the star is a CEMP-no; b)$A(Li)< 0.5$ is well belowthe plateau at $A(Li)=1.1$ found in the Lower Red Giant Branch stars,suggesting Li was originally depleted; and c)the isotopic ratio$^{12}$C/$^{13}$C is 87$\pm6$ showing very low $^{13}$C in contrast to whatexpected from a spinstar progenitor. We confirm that HE 0107$-$5240 is a binarystar with a long period of about 13000d ($\sim36$ years).The carbon isotopicratio excludes the possibility that the companion has gone through the AGBphase and transferred mass to the currently observed star. The binarity of HE0107$-$5240 implies some of the first generations of low-mass stars form inmultiple systems and indicates that the low metallicity does not preclude theformation of binaries. Finally, a solid indication of $v_{ rad}$ variation hasbeen found also in SMSS 1605$-$1443.

Espresso observations of HE 0107$-$5240 and other CEMP-no stars with $\rm [Fe/H]\le -4.5$