  Stationary axisymmetric binary configurations of unequal Kerr sources with a
massless strut among them are developed in a physical representation. In order
to describe interacting black holes, the axis conditions in the most general
case are solved analytically deriving the corresponding $5$-parametric
asymptotically flat exact solution. In addition, we obtain concise formulas for
the black hole horizons, the interaction force, as well as the thermodynamical
characteristics of each source in terms of physical Komar parameters: mass
$M_{i}$, angular momentum $J_{i}$, and coordinate distance $R$, where such
parameters are contained inside of the coefficients of a cubic equation which
can be interpreted as a dynamical law for interacting black holes with struts.
Some limits are obtained and discussed.
