On the transformation group of the second Painleve equation

We show that for the second Painleve equation y " = 2y(3) + ty + alpha, the Backlund transformation group G, which is isomorphic to the extended affine Weyl group of type (A) over cap(1), operates regularly on the natural projectification chi(c)/C(c, t) of the space of initial conditions, where c = alpha - 1/2. chi(c)/C(c, t) has a natural model chi[c]/C(t)[c]. The group G does not operate, however, regularly on chi[c]/C(t)[c]. To have a family of projective surfaces over C(t)[c] on which G operates regularly, we have to blow up the model chi[c] along the projective lines corresponding to the Riccati type solutions.