where each x_y is a distinct integer.

(Or prove that it is impossible).

Suppose a solution exists.  Note then that there are at least 7 odd integers among the {x_i}, as 1999 is 15 (mod 16) and as each perfect fourth power is 0 or 1 (mod 16) depending on whether it is even or odd, respectively.  But the largest of these 7 odd integers is at least as large as 7^4=2401>1999, a contradiction.

Thus no solution exists.