  Let $R^{n+1, n}$ be the vector space $R^{2n+1}$ equipped with the bilinear
form $(X,Y)=X^t C_n Y$ of index $n$, where $C_n= \sum_{i=1}^{2n+1} (-1)^{n+i-1}
e_{i, 2n+2-i}$. A smooth $\gamma: R\to R^{n+1,n}$ is {\it isotropic} if
$\gamma, \gamma_x, \ldots, \gamma_x^{(2n)}$ are linearly independent and the
span of $\gamma, \ldots, \gamma_x^{(n-1)}$ is isotropic. Given an isotropic
curve, we show that there is a unique up to translation parameter such that
$(\gamma_x^{(n)}, \gamma_x^{(n)})=1$ (we call such parameter the isotropic
parameter) and there also exists a natural moving frame. In this paper, we
consider two sequences of curve flows on the space of isotropic curves
parametrized by isotropic parameter. We show that differential invariants of
these isotropic curves satisfy Drinfeld-Sokolov's KdV type soliton hierarchies
associated to the affine Kac-Moody algebra $\hat B_n^{(1)}$ and $\hat
A_{2n}^{(2)}$ Then we use techniques from soliton theory to construct
bi-Hamiltonian structure, conservation laws, Backlund transformations and
permutability formulas for these curve flows.
