Fourier-integral operators and related mathematics

I would like to apply the branch of mathematics known as Fourier-integral operators (FIOs) to analyze equations of motion (5). In preparation for this, in 2003-2004 I took a reading course chaired (and with some lectures) by Len Bos and Andrej Bóna on functional analysis (Zimmer, 1990), partial differential equations (McOwen, 2003), pseudo-differential operators (Saint Raymond, 1991; Treves, 1980a; Wakabayashi, 2000), Fourier-integral operators (Treves, 1980b; Hörmander, 1971; Duistermaat and Hörmander, 1972; Wakabayashi, 2000) and the method of characteristics. After my completion of the standard method of characteristics and principal-symbol variant application to equations (5) I will move from the principal-symbol variant method to applying the full FIO method to equations (5).