We present an extensive study of the structural, magnetic and thermodynamic properties of high-quality monocrystals of the two heterometallic oxyborates from the ludwigite family: Cu$_2$GaBO$_5$ and Cu$_2$AlBO$_5$ in the temperature range above 2 K. The distinctive feature of the investigated structures is the selective distribution of Cu and Ga/Al cations. The unit cell of Cu$_2$GaBO$_5$ and Cu$_2$AlBO$_5$ contains four nonequivalent crystallographic sites of metal ions. Two sites in the structure from four nonequivalent crystallographic sites of metal ions of Cu$_2$GaBO$_5$ are fully occupied by Cu atoms which form the quasi one-dimensional chains along the a-axis. For Cu$_2$AlBO$_5$ all sites are partially occupied by Al and Cu atoms. The joint analysis of low-temperature data on magnetic susceptibility and magnetic contribution to the specific heat showed that Cu$_2$AlBO$_5$ and Cu$_2$GaBO$_5$ exhibit an antiferromagnetic transition at $T_{rm N} approx 3$ and 4 K, respectively. The magnetic contributions to the specific heat for both compounds were obtained after subtracting the phonon contribution. It is shown that the external magnetic field above 2.5 T leads to a broadening of the magnetic phase transition indicating suppression of the long-range antiferromagnetic order.
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