n ....................... 150. 2.4. The KP vector fields ........................ 152. ix 3. Multi-Hamiltonian structure and symmetries ................ 155. 3.1. The loop algebra 9(q ........................ 155. 3.2. Reducing the R-brackets and the vector field ~ ............. 157. 4. The odd and the even Mumford systems .................. 161. 4.1. The (odd) Mumford system ..................... 161. 4.2. The even Mumford system ...................... 163.