  We revisit the Hitchin integrable system whose phase space is the bundle
cotangent to the moduli space $N$ of holomorphic $SL_2$-bundles over a smooth
complex curve of genus two. $N$ may be identified with the 3-dimensional
projective space of theta functions of the second order, We prove that the
Hitchin system on $T^*N$ possesses a remarkable symmetry: it is invariant under
the interchange of positions and momenta. This property allows to complete the
work of van Geemen-Previato which, basing on the classical results on geometry
of the Kummer quartic surfaces, specified the explicit form of the Hamiltonians
of the Hitchin system. The resulting integrable system resembles the classic
Neumann systems which are also self-dual. Its quantization produces a commuting
family of differential operators of the second order acting on homogeneous
polynomials in four complex variables. As recently shown by van Geemen-de Jong,
these operators realize the Knizhnik-Zamolodchikov-Bernard-Hitchin connection
for group SU(2) and genus 2 curves.
