  This is the sixth in a series of papers constructing examples of special
Lagrangian m-folds in C^m. We present a construction of special Lagrangian
cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers
of the series. Then we generalize it to a construction of non-conical special
Lagrangian 3-folds in C^3 involving three commuting o.d.e.s.
  Now special Lagrangian cones in C^3 are linked to the theory of harmonic maps
and integrable systems. Harmonic maps from a Riemann surface into complex
projective space CP^n are an integrable system, and can be studied and
classified using loop group techniques. If N is a special Lagrangian cone in
C^3, then N is the cone on the image of a conformal harmonic map \psi : S -->
S^5 for some Riemann surface S, and the projection of \psi to CP^2 is also
conformal harmonic.
  Our examples of special Lagrangian cones in C^3 yield conformal harmonic maps
\psi : R^2 --> CP^2. We work through the integrable systems theory for these
examples, showing that they are superconformal of finite type, and calculating
their harmonic sequences, Toda and Tzitzeica solutions, algebra of polynomial
Killing fields and spectral curves. We also study the double periodicity
conditions for \psi, and so find families of superconformal tori in CP^2.
  We finish by asking whether our more general construction of special
Lagrangian 3-folds can also be derived from a higher-dimensional integrable
system, and whether the special Lagrangian equations themselves are in some
sense integrable.
