  The Calogero-Painlev\'e systems were introduced in 2001 by K. Takasaki as a
natural generalization of the classical Painlev\'e equations to the case of the
several Painlev\'e "particles" coupled via the Calogero type interactions. In
2014, I. Rumanov discovered a remarkable fact that a particular case of the
second Calogero-Painlev\'e II equation describes the Tracy-Widom distribution
function for the general beta-ensembles with the even values of parameter beta.
Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it was
proven that all Calogero-Painlev\'e systems are Lax integrable, and hence their
solutions admit a Riemann-Hilbert representation. This important observation
has opened the door to rigorous asymptotic analysis of the Calogero-Painlev\'e
equations which in turn yields the possibility of rigorous evaluation of the
asymptotic behavior of the Tracy-Widom distributions for the values of beta
beyond the classical $\beta =1, 2, 4.$ In this work we shall start an
asymptotic analysis of the Calogero-Painlev\'e system with a special focus on
the Calogero-Painlev\'e system corresponding to $\beta = 6$ Tracy-Widom
distribution function.
  Some technical details were omitted here and will be presented in the next
version of the text.
