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% \usepackage{equation}

\externaldocument[A-]{main}

\begin{document}

\section{File 1Introduction}
\label{sec:introduction}

(\ref{A-eq:main:1})
This is the external file

New test
\begin{equation}
  \label{eq:file1:1}
  \int f dx
\end{equation}

\begin{subequations}
  \label{E:EULERPOISSON}
  \begin{alignat}{2}
    \partial_t\rho + \text{div}\, (\rho \mathbf{u})& = 0 &&\ \text{ in } \ B(t)\,;\label{E:CONTINUITYEP}\\
    \rho\left(\partial_t  \mathbf{u}+ ( \mathbf{u}\cdot\nabla) \mathbf{u}\right) +\nabla p &=-\rho \nabla\Phi&&\ \text{ in } \ B(t)\,;\label{E:VELOCITYEP}\\
    \Delta \Phi  = 4\pi c\,\rho, \ \lim_{|x|\to\infty}\Phi(t,x) & = 0&& \ \text{ in } \ {\mathbb R}^3 \,; \label{E:POISSON}\\
    %\lim_{|x|\to\infty}\Phi(t,x) & = 0; && \label{E:POISSONBOUNDARY}\\
    p&=0&& \ \text{ on } \ \partial B(t) \,;\label{E:VACUUMEP} \\
    \mathcal{V}_{\partial B(t)}&= \mathbf{u}\cdot \mathbf{n}(t)  && \ \text{ on } \ \partial B(t)\,;\label{E:VELOCITYBDRYEP}\\
    (\rho(0,\cdot),  \mathbf{u}(0,\cdot))=(\rho_0,  \mathbf{u}_0)\,, & \ B(0)=B_0&&\,.\label{E:INITIALEP}
  \end{alignat}
\end{subequations}


Now let's reference the eq.~(\ref{eq:file1:1}).

\end{document}

