A number of properties of Toeplitz operators with 2 × 2 symbol G canbe studied from the solutions to an associated Riemann-Hilbert problem(RHP) G\phi_+ = \phi_ , where \phi_{±} belong to certain spaces of analytic functions in C± . We present here an approach to determining solutions to a RHP ofthat type, with almost periodic coefficient G, based on the so-called tablemethod presented for the first time in [1]. This allows to establish necessaryand sufficient conditions for the invertibility of the Toeplitz operator andto determine its inverse explicitly. Some unexpected, but interesting resultsregarding the Fourier spectrum of the solutions are obtained which are notapparent through other approaches to the same problem.

[1] Camara, M. C.; dos Santos, A. F.; Martins, M. C., A new approach tofactorization of a class of almost-periodic triangular symbols and relatedRiemann-Hilbert problems, J. Funct. Anal. 235 (2006), no. 2, 559 - 592.