**mixed fraction**with the respective fractional parts of x and y equal to 1/3 and 2/3 and, z is a positive integer such that:

[x

^{2}] + [y

^{2}] = z

^{2}, and:

x + y is a perfect square

(I) Determine the pairs (x, y) that generate the first five smallest values of z.

(II) Determine the smallest common value of z, such that there are precisely two pairs (x, y) that satisfy all the given conditions. What are the next two smallest values of z with this property?

__Note__: [N] denotes the greatest integer ≤ N