Specialists are now convinced that reliable storage of radioactive wastes with half-lives of T1/2 -30 yr, such as, 137Cs and 9~ produced in the nuclear power production is possible. However, it is much more difficult to prove the safety of the storage of longer-lived radiologically toxic nuclei, for example, 129I (T1/2 -1.5.107 yr) and 237Np (T1/2 -2.106 yr). This is why specialists return again and again to the idea of transmutation of such nuclei into short-lived nuclei. This problem will become more acute in the future, since the amount of wastes produced in nuclear power production worldwide is still far from equilibrium. It is also obvious that until comparative investigations of these approaches are made, it Will be impossible to make a final choice of the best approach or to determine whether or not any advantage is gained by combining different approaches.There are many technological and economic limitations that make it impossible to burn up in standard reactors with a solid-fuel core all nuclear wastes that cannot be stored. For this reason, a new type of reactor is required to switch to a closed fuel cycle: a reactor whose main properties are the ability to burn upall wastes, the possibility of flexible control of the nuclide composition, and enhanced safety. The existence of such a reactor will give the closed fuel cycle the required efficiency and flexibility.From our point of view, one such reactor is a subcritical liquid-salt reactor that is replenished by an external neutron source [1-10]. Its positive qualities are the possibility of obtaining a tow quantity of fission fragments in the core, which makes it possible to decrease substantially the residual heat release, and the possibility of removing heat by natural circulation of fused salt and a high negative temperature coefficient of the reactivity. These and omer qualities of the reactor eliminate accidents associated with the disruption of heat removal.The motives for studying a subcritical reactor are obvious: adequate subcriticality eliminates reactivityaccident and after the external neutron source is switched off the reactor stops within -10 -3 sec, i.e. the external source is an additional instantaneous means of control. Systems based on magnetic confinement of plasma (tokamaks) [11,12], laser thermonuclear fusion [13], muon catalysis [14,15], gas-dynamic plasma traps [16], and so on have been proposed as external sources of neutrons. At the present time the possibility of using as such sources accelerators which accelerate protons up to -1 GeV are being actively discussed [17][18][19].Our objectives in the present paper, which is a continuation of previous investigations [6][7][8][9][10], are as follows: selection of an optimal scheme for a safe subcritical reactor, which would make it possible to achieve high neutron multiplication while at the same time eliminating reactivity accidents, to decrease substantially the intensity of the external neutron source and thereby to eliminate the main drawbacks of the standard proton-beam schemes...