**Background:** The equation $$a^4+b^4+c^4=2d^4$$ has infinitely many positive integral solutions if we take $c=a+b$ and $a^2+a\cdot b+b^2=d^2$. **Main problem:** Find some positive integral solutions to the equation $$a^4+b^4+c^4=2d^4$$ with $a\lt b\lt c\ne a+b$ and $GCD(a,b,c)=1$.