Diophantine equations with sum of cubes and cube of sum

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 $a\mathrm{\ne }0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any cube 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 $1/a = 1- 24/m$withrestrictionsontheinteger$m$. The equations can be represented by elliptic curves unless $a=9$ or 1, and any elliptic curve of nonzero $j$-invariant and torsion group $\mathbb{Z}\mathrm{/}3k\mathbb{Z}$ for $k = 2,3,4$,or$$\mathbb{Z}$/2$\mathbb{Z}$ \times $\mathbb{Z}$/6$\mathbb{Z}$ $ corresponds to a particular $a$. We prove that for any $a$ the number of nontrivial solutions is at most 3 or is infinite, and for integer $a$ it is either 0 or $\mathrm{\infty }$. For $a=9$, we find the general solution, 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.