The generalized Moyal-Nahm and continuous Moyal-Toda equations

We present in detail a class of solutions to the 4D SU(infinity) Moyal-anti-self-dual Yang-Mills (ASDYM) equations (an effective 6D theory) that are related to reductions of the generalized Moyal-Nahm equations using the Ivanova-Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitrary gauge groups in four dimensions. A further dimensional reduction of the above effective 6D equations yields solutions to the Moyal-anti-self-dual gravitational equations in four dimensions. The self-dual Yang-Mills/self-dual gravity case requires a separate study. The SU(2) Toda lattice and SU(infinity) (continuous) Moyal-Toda lattice equations are derived from the Moyal-Nahm equations. An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction of the Moyal heavenly equations) into the SU(2) Toda lattice field is found. Finally, the generalized Moyal-Nahm equations are conjectured that contain the (continuous) SU(infinity) Moyal-Toda lattice equations, after a suitable reduction process. Embeddings of the different types of Moyal-Toda lattice equations into the Moyal-Nahm equations are described. (C) 1999 American Institute of Physics. [S0022-2488(99)00508-3].