  In the framework of the quasipotential method in quantum electrodynamics we
calculate corrections to the nuclear structure proportional to $r_N^2$ from
two-photon exchange amplitudes in the fine and hyperfine structure of P-states
in muonic hydrogen, as well as the photon-photon interaction amplitudes,
leading to the exchange of the axial vector meson. In constructing the
quasipotential of the muon-nucleus interaction, we use the method of projection
operators on states of two particles with a definite spin and total angular
momentum. Analytical calculation of the matrix elements is performed and
contributions to the fine and hyperfine structure of the $2P_{1/2}$ and
$2P_{3/2}$ levels are obtained.
