  We study the Tracy-Widom distribution function for Dyson's $\beta$-ensemble
with $\beta = 6$. The starting point of our analysis is the recent work of I.
Rumanov where he produces a Lax-pair representation for the Bloemendal-Vir\'ag
equation. The latter is a linear PDE which describes the Tracy-Widom functions
corresponding to general values of $\beta$. Using his Lax pair, Rumanov derives
an explicit formula for the Tracy-Widom $\beta=6$ function in terms of the
second Painlev\'e transcendent and the solution of an auxiliary ODE. Rumanov
also shows that this formula allows him to derive formally the asymptotic
expansion of the Tracy-Widom function. Our goal is to make Rumanov's approach
and hence the asymptotic analysis it provides rigorous. In this paper, the
first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a
certain gauge transformation of the standard Lax pair for the second Painlev\'e
equation. This gauge transformation though contains functional parameters which
are defined via some auxiliary nonlinear ODE which is equivalent to the
auxiliary ODE of Rumanov's formula. The gauge-interpretation of Rumanov's
Lax-pair allows us to highlight the steps of the original Rumanov's method
which needs rigorous justifications in order to make the method complete. We
provide a rigorous justification of one of these steps. Namely, we prove that
the Painlev\'e function involved in Rumanov's formula is indeed, as it has been
suggested by Rumanov, the Hastings-McLeod solution of the second Painlev\'e
equation. The key issue which we also discuss and which is still open is the
question of integrability of the auxiliary ODE in Rumanov's formula. We note
that this question is crucial for the rigorous asymptotic analysis of the
Tracy-Widom function.
