Integrability of Riccati equations and the stationary KdV equations

Using the S.Lie’s infinitesimal approach we establish the connection between integrability of a one-parameter family of the Riccati equations and the stationary KdV hierarchy. In this paper we will suggest a method for integrating a one-parameter family of the Riccati equations ux + u 2 = f(x, λ) (1) based on their Lie symmetries. Here f(x, λ) = λ + λVn−1(x) + · · ·+ λV1(x) + V0(x) and λ is an arbitrary real parameter. We recall the principal idea of application of Lie group methods to integrating the Riccati equation (1). Suppose it admits a one-parameter transformation group having the infinitesimal operator X = ξ(x, u, λ) ∂ ∂x + η(x, u, λ) ∂ ∂u . Then making a change of variables (x, u) → (x̃, ũ) transforming X to become X̃ = ∂ ∂x̃ (which is always possible) reduces the equation under study ∗e-mail: [email protected]