  Since 1772, when Euler first described two methods of obtaining two pairs of
biquadrates with equal sums, several methods of solving the diophantine
equation $x^4+y^4=z^4+w^4$ have been published. All these methods yield
parametric solutions in terms of homogeneous bivariate polynomials of odd
degrees. In this paper we describe a method that yields three parametric
solutions of the aforesaid diophantine equation in terms of homogeneous
bivariate polynomials of even degrees, namely degrees~$74$, $88$ and $132$
respectively.
