  We apply one of the exactly symplectic integrators, that we call HB15, of
\cite{HB15}, along with the Kepler problem solver of \cite{WH15}, to solve
planetary system $N$-body problems. We compare the method to Wisdom-Holman
methods (WH) in the \texttt{MERCURY} software package, the \texttt{MERCURY}
switching integrator, and others and find HB15 to be the most efficient method
or tied for the most efficient method in many cases. Unlike WH, HB15 solved
$N$-body problems exhibiting close encounters with small, acceptable error,
although frequent encounters slowed the code. Switching maps like
\texttt{MERCURY} change between two methods and are not exactly symplectic. We
carry out careful tests on their properties and suggest they must be used with
caution. We then use different integrators to solve a 3-body problem consisting
of a binary planet orbiting a star. For all tested tolerances and time steps,
\texttt{MERCURY} unbinds the binary after 0 to 25 years. However, in the
solutions of HB15, a time-symmetric Hermite code, and a symplectic Yoshida
method, the binary remains bound for $>1000$ years. The methods' solutions are
qualitatively different, despite small errors in the first integrals in most
cases. Several checks suggest the qualitative binary behavior of HB15's
solution is correct. The Bulirsch-Stoer and Radau methods in the
\texttt{MERCURY} package also unbind the binary before a time of 50 years,
suggesting this dynamical error is due to a \texttt{MERCURY} bug.
