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In Part I (arXiv:1209.2045) we computed the Stokes data, though not the"connection matrix", for the smooth solutions of the tt*-Toda equations whoseexistence we established by p.d.e. methods. Here we give an alternative proofof the existence of some of these solutions by solving a Riemann-Hilbertproblem. In the process, we compute the connection matrix for all smoothsolutions, thus completing the computation of the monodromy data. We also giveconnection formulae relating the asymptotics at zero and infinity of all smoothsolutions, clarifying the region of validity of the formulae establishedearlier by Tracy and Widom. Finally, for the tt*-Toda equations, we resolvesome conjectures of Cecotti and Vafa concerning the positivity of S+S^t (whereS is the Stokes matrix) and the unimodularity of the eigenvalues of themonodromy matrix.

Isomonodromy aspects of the tt* equations of Cecotti and Vafa II. Riemann-Hilbert problem