Ambiguous solutions of a Pell equation

It is known that if the negative Pell equation X2 - DY2 = -1 is solvable (in integers), and if (x, y) is its solution with the smallest positive x and y, then all of its solutions (xn, yn) are given by the formula root root xn + yn D = +/-(x + y D)2n+1 for n is an element of 71. Furthermore, a theorem of Walker from 1967 states that if the equation a X2 -bY2 = +/- 1 is solvable, and if (x, y) is its solution with the smallest positive x and y, then all of its solutions (xn, yn) are given by root xn root a + yn root b = +/-(x root a + y b)2n+1 for n is an element of 71. We prove a unifying theorem that includes both of these results as spe-cial cases. The key observation is a structural theorem for the nontrivial ambigu-ous classes of the solutions of the (generalized) Pell equations X2 - DY2 = +/- N. We also provide a criterion for determination of the nontrivial ambiguous classes of the solutions of Pell equations.