== General N-body systems ==
The Hermite integration scheme (by far the most commonly used scheme
for dense clusters' dynamical evolution) is described here:
http://adsabs.harvard.edu/abs/1992PASJ...44..141M
This paper also describes a common technique for handling
density differences in a cluster efficiently.

Various schemes for determining the timestep is here:
http://adsabs.harvard.edu/abs/1991ApJ...369..200M

A generalization of the Hermite scheme to 6th and 8th orders is in:
http://adsabs.harvard.edu/abs/2008NewA...13..498N

It may look like adaptive timesteps are not good for long term
integrations, but some of the long term properties of symplectic
integrators are recovered by time symmetrization, i.e., letting
the length of the timestep depend on the values of the variables at
the beginning and at the end of the step:
http://adsabs.harvard.edu/abs/2006NewA...12..124M

An example of mixing two ways of calculating forces via splitting
methods is in
http://adsabs.harvard.edu/abs/2007PASJ...59.1095F

Finally most of these are explained in a more pedagogical manner at
http://www.artcompsci.org/
This site also has a discussion of initial conditions.

== Near-Keplerian N-body systems ==
The most commonly used code, Mercury, is described in the paper:
http://adsabs.harvard.edu/abs/1999MNRAS.304..793C
This paper discusses the 3-way splitting I mentioned. This
splitting was discovered by
http://adsabs.harvard.edu/abs/1998AJ....116.2067D
but their treatment of close encounters is too complicated for
my taste.

The problem arising from the close encounters with the central
object (nearly parabolic orbits, or orbits with eccentricity nearly
unity, or orbits with almost zero energy in the Keplerian part of the
Hamiltonian) is described and to some extent solved in
http://adsabs.harvard.edu/abs/2000AJ....120.2117L

An interesting integrator that uses splitting, but is not symplectic
is given in
http://adsabs.harvard.edu/abs/2008MNRAS.384..323L

== Further Thoughts ==
A common bottleneck in near-Keplerian integrators is solving Kepler's
equation for arbitrary epoch. The ``standard'' approach is described
in Danby's book:
http://adsabs.harvard.edu/abs/1992fcm..book.....D
He also has a series of three papers:
http://adsabs.harvard.edu/abs/1983CeMec..31...95D
http://adsabs.harvard.edu/abs/1983CeMec..31..317B
http://adsabs.harvard.edu/abs/1987CeMec..40..303D
I have the feeling that there is room for improvement here.

I implemented a Kepler solver in C. The code and the
ideas I used can be found in
http://www.strw.leidenuniv.nl/~gurkan/kepler/sol_kep/sol_kep.html

The implementation of Kepler solver used by Mercury is part of
the SWIFT package:
http://www.boulder.swri.edu/~hal/swift.html

Even though I did not mention in my talk, an important technique
in few body problems used in connection to splitting methods is
the time transformation, in particular "algorithmic regularization".
This is of course well known in applied mathematics community as
well (e.g., Blanes & Budd's EASY integrators). Here are a few
references by astronomers:
http://adsabs.harvard.edu/abs/1997CeMDA..67..145M
http://adsabs.harvard.edu/abs/1999CeMDA..74..287M
http://adsabs.harvard.edu/abs/1999AJ....118.2532P

Seppo Mikkola has done a lot of work related to few-body integrations:
http://adsabs.harvard.edu/abs/2002CeMDA..82..375M
http://adsabs.harvard.edu/abs/2009NewA...14..607H

The first paper (and references therein) discusses transforming back
and forth, and hence sidestepping solving Kepler's equation by
iteration. The second one demonstrates that an important perturbation
(1/r^2) can be integrated analytically. I have yet to implement
either of these.