  HE 0107$-$5240 is a hyper metal-poor star with $\rm [Fe/H]=-5.39$. We
performed high-res observations with the ESPRESSO spectrograph at the VLT to
constrain the kinematical properties of the binary system HE 0107$-$5240 and to
probe 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 the
interval 4200$-$4315A, which contains the strong CH band, against a template in
an iterative process. A Bayesian method is applied to calculate the orbit by
using the ESPRESSO measurements and others from the literature. A chemical
analysis has also been performed in HE0107$-$5240 by means of spectral
synthesis. Observations of HE 0107$-$5240 spanning more than 3 years show a
monotonic decreasing trend in radial velocity at a rate of approximately by 0.5
ms$^{-1}$d$^{-1}$. The period is constrained at $P_{\rm orb} =
13009_{-1370}^{+1496}$d. New more stringent upper-limits have been found for
several 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 below
the 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 what
expected from a spinstar progenitor. We confirm that HE 0107$-$5240 is a binary
star with a long period of about 13000d ($\sim36$ years).The carbon isotopic
ratio excludes the possibility that the companion has gone through the AGB
phase and transferred mass to the currently observed star. The binarity of HE
0107$-$5240 implies some of the first generations of low-mass stars form in
multiple systems and indicates that the low metallicity does not preclude the
formation of binaries. Finally, a solid indication of $v_{ rad}$ variation has
been found also in SMSS 1605$-$1443.
