How can I spread the ToC over two slides in beamer?

Question

As I have a rather long Table of Contents in my beamer document (I have four section, each composed of at least five subsections), I'd like to spread it over two slides rather than just one. How can I do that? My code so far:

\documentclass[leqno,mathserif]{beamer}
\usepackage{graphics,bm}
\usetheme{Luebeck}
\setbeamercovered{transparent}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english,french]{babel}

\newcommand{\diff}{\mathop{}\mathopen{}\mathrm{d}}
\setbeamercovered{transparent}
\usepackage{enumitem}
%the university of bamberg colors
% colors to be used
\definecolor{fu-blue}{RGB}{0, 51, 102} % blue text
\definecolor{fu-green}{RGB}{153, 204, 0} % green text
\definecolor{fu-red}{RGB}{204, 0, 0} % red text (used by \alert)


%the usage of the colors in special elements
%\setbeamercolor{title}{fg=UBBlack}
%\setbeamercolor{frametitle}{UBBlack}
\setbeamercolor{structure}{fg=fu-blue}

\usepackage[tightpage]{preview}


\setbeamercolor{block title example}{fg=black,bg=red!20!green!50!white}
\setbeamercolor{block title}{fg=black,bg=blue!50!white}
\setbeamercolor{block title alerted}{fg=black,bg=red!70!white}


\AtBeginSection[] 
{ 
\begin{frame}<beamer>{Plan} 
\tableofcontents[currentsection] 
\end{frame} 
} 

\title[SDE]{SDE}
\subtitle{XXXXXXXXXXXXXXXXXXX}
\author[XXXXXXXXXXXX]{XXXXXXX} 
\institute{XXXXXXXX}
\date{\today}

%\setbeamertemplate{navigation symbols}{\insertslidenavigationsymbol}

\begin{document}
\begin{frame}
\titlepage
\end{frame}
\section[Plan]{}
\begin{frame}
\tableofcontents
\end{frame}

\section[Introduction]{Introduction}
\begin{frame}
(introduction)
\end{frame}

\section[EDS]{EDS}
\subsection[Définition XXX]{Définition XXX}
\begin{frame}
\frametitle{Définition XXXX}
bla bla bla bla 
\end{frame}

\subsection[Solution XXXXXXX]{Solution XXX}
\begin{frame}
\frametitle{Solution XXXX}
bla bla bla bla bla
\end{frame}

\subsubsection[Diverses notions de solutions]{Diverses notions de solutions}
\begin{frame}
\frametitle{Diverses XXXXXXXXXXX}
\framesubtitle{Solution XXXXXXX}
bla bla bla
\end{frame}

\begin{frame}
\frametitle{Diverses XXXXX}
\framesubtitle{Solution XXXXX}
bla bla bla
\end{frame}
\subsubsection{Diverses XXXX}
\begin{frame}
\frametitle{Diverses XXXX}
\framesubtitle{Unicité }
bla bla bla 

\end{frame}

\subsection{Diverses notions d'existence et unicité des solutions }
\begin{frame}
\frametitle{Existence et unicité des solutions }
\framesubtitle{Sous Condition de lipschitzienne }
bla bla bla
\end{frame}

\begin{frame}
\frametitle{Existence et unicité des solutions}
\framesubtitle{Sous Condition de XXXXXXX}
\end{frame}


\section[Diffusion]{Diffusion}
\subsection{Définition d'une diffusion d'Itô}
\begin{frame}
\frametitle{Définition d'une diffusion d'Itô}
\framesubtitle{Notation}
bla bla bla 
\end{frame}

\begin{frame}
\frametitle{Définition d'une diffusion d'Itô}
bla bla bla 
\end{frame}

\subsection{Flot stochastique}
\begin{frame}
\frametitle{Flot stochastique}
\framesubtitle{propriété du Flot stochastique}
bla bla bla 
\end{frame}

\subsection{La propriété de Markov}
\begin{frame}
\frametitle{La propriété de Markov}
\framesubtitle{Notation}
bla bla bla 
\end{frame}

\begin{frame}
\frametitle{Propriété de Markov}
\framesubtitle{Propriété de Markov version :Faible et Forte}
bla bla 
\end{frame}

\subsection{Semi-groupes et générateurs d'une diffusion d'Itô}
\begin{frame}
\frametitle{Semi-groupes}
\frametitle{Définition d'un Semi-groupes}
bla bla bla 
\end{frame}

\begin{frame}
\frametitle{Définition d'un Générateur d'une diffusion d'Itô}
\framesubtitle{Notation}
bla bla 
\end{frame}

\subsection{La formule de Dynkin}
\begin{frame}
bla bla bla 
\end{frame}

\section[Simulation]{Simulation}
\subsection{A-1}
\begin{frame}
bla bla bla 
\end{frame}
\subsection{A-2}
\begin{frame}
bla bla bla
\end{frame}
\subsection{A-3}
\begin{frame}
bla bla
\end{frame}
\subsection{A-4}
\begin{frame}
bla bla 
\end{frame}
\subsection{A-5}
\begin{frame}
bla bla
\end{frame}
\end{document}

Answer 1

You can use the allowframebreaks option. It tells beamer that if the contents do not fit in the current slide it can put the overflowing lines in a separate slide (or more, as needed). It is very handy for table of contents and bibliographies.

In your case just put

\begin{frame}[allowframebreaks]
    \tableofcontents
\end{frame}

If you want to apply this option to the frames generated by

\AtBeginSection[] 
{ 
\begin{frame}<beamer>{Plan} 
\tableofcontents[currentsection] 
\end{frame} 
}

you have to remove the overlay as allowframebreaks is not compatible with overlays. You can achieve the same effect with

\mode<presentation>{
    \AtBeginSection[] 
    { 
    \begin{frame}[allowframebreaks]{Plan} 
    \tableofcontents[currentsection] 
    \end{frame} 
    } 
}

However I do not recommend breaking the toc at every section. (Actually in my opinion it may be not shown at all, but that's another story)