  We introduce a new integrable equation valued on a Cayley-Dickson (C-D)
algebra. In the particular case in which the algebra reduces to the complex one
the new interacting term in the equation cancells and the equation becomes the
known Korteweg-de Vries equation. For each C-D algebra the equation has an
infinite sequence of local conserved quantities. We obtain a B\"{a}cklund
transformation in the sense of Walhquist-Estabrook for the equation for any
Cayley-Dickson algebra, and relate it to a generalized Gardner equation. From
it, the infinite sequence of conserved quantities follows directly. We give the
explicit expression for the first few of them. From the B\"{a}cklund
transformation we get the Lax pair and the one-soliton and two-soliton
solutions generalizing the known solutions for the quaternion valued KdV
equation. From the Gardner equation we obtain the generalized modified KdV
equation which also has an infinite sequence of conserved quantities. The new
integrable equation is preserved under a subgroup of the automorphisms of the
C-D algebra. In the particular case of the algebra of octonions, the equation
is invariant under $SU(3)$.
