= Solution
Treat $B_\mu$ and $B_\mu^*$ as independent variables in the <Euler-Lagrange equation>. Varying $B_\nu^*$ gives the <Proca equation>
$$
\partial_\mu G^{\mu\nu}+M^2B^\nu=0.
$$
The divergence of its first term vanishes by antisymmetry and commuting partial derivatives. Since $M\ne0$, it follows that $\partial_\nu B^\nu=0$. Substitution into the original equation gives
$$
\boxed{(\Box+M^2)B^\nu=0,\qquad\partial_\nu B^\nu=0.}
$$
The complex-conjugate variation gives $\partial_\mu G^{*\mu\nu}+M^2B^{*\nu}=0$, hence $(\Box+M^2)B^{*\nu}=0$ and $\partial_\nu B^{*\nu}=0$.
For an on-shell <plane wave> $B^\mu=\varepsilon^\mu e^{-ipx}$, the <Lorenz constraint in Proca theory> is $p_\mu\varepsilon^\mu=0$. In the rest frame it sets $\varepsilon^0=0$, leaving three complex <particle polarizations>. Thus the complex <Proca field> has \b[three independent complex dynamical components], or six real ones; its quantized particle and antiparticle each have three physical <particle polarizations>. The conjugate field is not another unrelated classical field.
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