  We describe an algorithmic reduction of the search for integral points on a
curve y^2 = ax^4 + bx^2 + c with nonzero ac(b^2-4ac) to solving a finite number
of Thue equations. While existence of such reduction is anticipated from
arguments of algebraic number theory, our algorithm is elementary and to best
of our knowledge is the first published algorithm of this kind. In combination
with other methods and powered by existing software Thue equations solvers, it
allows one to efficiently compute integral points on biquadratic curves.
  We illustrate this approach with a particular application of finding
near-multiples of squares in Lucas sequences. As an example, we establish that
among Fibonacci numbers only 2 and 34 are of the form 2m^2+2; only 1, 13, and
1597 are of the form m^2-3; and so on.
  As an auxiliary result, we also give an algorithm for solving a Diophantine
equation k^2 = f(m,n)/g(m,n) in integers m,n,k, where f and g are homogeneous
quadratic polynomials.
