  This paper is the first in a forthcoming series of works where the authors
study the global asymptotic behavior of the radial solutions of the 2D periodic
Toda equation of type $A_n$. The principal issue is the connection formulae
between the asymptotic parameters describing the behavior of the general
solution at zero and infinity. To reach this goal we are using a fusion of the
PDE analysis and the Riemann-Hilbert nonlinear steepest descent method of Deift
and Zhou which is applicable to 2D Toda in view of its Lax integrability. A
principal technical challenge is the extension of the nonlinear steepest
descent analysis to Riemann-Hilbert problems of matrix rank greater than $2$.
In this paper, we meet this challenge for the case $n=2$ (the rank $3$ case)
and it already captures the principal features of the general $n$ case.
