On $a^4+b^4+c^4+d^4=(a+27(b+c+d){)}^4$ and ten other new infinite families

In 2008, Jacobi and Madden proved that $a^4+b^4+c^4+d^4=e^4$ has infinitely many primitive solutions with the linear relation $e=a+b+c+d$. We extend their result to relations with rational cubes $e=a+k^3(b+c+d)$ and prove infinitude for $k=3$ and $10$ other $k\mathrm{\ne }1$. We compile $30$ starting solutions. In the spirit of Jacobi and Madden’s original paper, everything here is simple algebra except one theorem of Mazur.