  We solve Diophantine equations of the type $ \, a \, (x^3 + y^3 + z^3 ) = (x
+ y + z)^3$, where $x,y,z$ are integer variables, and the coefficient $a \neq
0$ is rational. We show that there are infinite families of such equations,
including those where $a$ is any ratio of cubes or certain rational fractions,
that have nontrivial solutions. There are also infinite families of equations
that do not have any nontrivial solution, including those where $1/a = 1 -
24/m$ with certain restrictions on the integer $m$. The equations can be
represented by elliptic curves unless $a = 9$ or 1. If $a$ is an integer and
two variables are equal and nonzero, there exist nontrivial solutions only for
$a=4$ or 9; there are no solutions for $a = 4$ when $xyz \neq 0$. Without
imposing constraints on the variables, we find the general solution for $a =
9$, which depends on two integer parameters. These cubic equations are
important in particle physics, because they determine the fermion charges under
the $U(1)$ gauge group.
