  We consider classical superstrings propagating on AdS_5 x S^5 space-time. We
consistently truncate the superstring equations of motion to the so-called
su(1|1) sector. By fixing the uniform gauge we show that physical excitations
in this sector are described by two complex fermionic degrees of freedom and we
obtain the corresponding Lagrangian. Remarkably, this Lagrangian can be cast in
a two-dimensional Lorentz-invariant form. The kinetic part of the Lagrangian
induces a non-trivial Poisson structure while the Hamiltonian is just the one
of the massive Dirac fermion. We find a change of variables which brings the
Poisson structure to the canonical form but makes the Hamiltonian nontrivial.
The Hamiltonian is derived as an exact function of two parameters: the total
S^5 angular momentum J and string tension \lambda; it is a polynomial in 1/J
and in \sqrt{\lambda'} where \lambda'=\frac{\lambda}{J^2} is the effective BMN
coupling. We identify the string states dual to the gauge theory operators from
the closed su(1|1) sector of N=4 SYM and show that the corresponding near-plane
wave energy shift computed from our Hamiltonian perfectly agrees with that
recently found in the literature. Finally we show that the Hamiltonian is
integrable by explicitly constructing the corresponding Lax representation.
