We consider classical superstrings propagating on AdS₅ × S⁵ space-time. We consistently truncate the superstring equations of motion to the so-called (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 S⁵ angular momentum J and string tension λ; it is a polynomial in 1/J and in (λ')^1/2 where λ' = λ/J² is the effective BMN coupling. We identify the string states dual to the gauge theory operators from the closed (1|1) sector of = 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.