On Toda equation and half BPS supergravity solution in M-theory

Recently, it was shown that half BPS Supergravity solution of theories withSU(2$|$4) symmetry algebra is given uniformly by determining a single functionwhich obeys three dimensional continuous Toda equation. In this paper, we studythe scale invariant solution of Toda equation. Our motivation is that somesolutions of half BPS sector of IIB supergravity, as one excepts from thefermion description of the theory, are scale invariant. By defining twoauxiliary functions we prove that such solutions of Toda equation obey cubicalgebraic equation. We obtain some simpl solutions of Toda equation specially,we observe that the PP-wave solution can be written in this fashion.Comment: 13 pages, no figure