  In this paper we consider Diophantine equation x4 + y4 = z4 + w4 (1)We
construct some family of cubic curves.We prove that every rational point on
Quar- tica x4 + y4 = z4 + w4 can be mapped to a point on some curve of this
family. We also prove the opposite: each rational point belonging to our family
of curves can be mapped to a rational point on the Quartica. (2) We find the
point on our family of curves corresponding to a parametric solution of Leonard
Euler. We construct several new parametric solutions of our Quartica, using a
parametric solution of Leonard Euler and the algebraic operation on the cubic
curves. (3)We present an algorithm to find all rational points on our Quartica.
