= Solution
For an occupied uniform <two-component Bose-Einstein condensate>, the <gradient> and trapping terms vanish. Dividing each stationary equation by its nonzero field gives
$$
\boxed{\mu_1=U_{11}n_1+U_{12}n_2,\qquad\mu_2=U_{12}n_1+U_{22}n_2.}
$$
The same relations follow by differentiating the uniform interaction-energy density $e=U_{11}n_1^2/2+U_{22}n_2^2/2+U_{12}n_1n_2$ with respect to each <number density>. When $D=U_{11}U_{22}-U_{12}^2\ne0$, the inverse relations are
$$
n_1=\frac{U_{22}\mu_1-U_{12}\mu_2}{D},\qquad
n_2=\frac{U_{11}\mu_2-U_{12}\mu_1}{D}.
$$
These describe a genuinely overlapping mixture only when both densities are nonnegative. An unoccupied component cannot be divided out of its stationary field equation; it requires the appropriate constrained equilibrium condition instead.
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