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Using the Doppler-free, perturbation-facilitated optical-optical double resonance technique, we have investigated the vibrational, rotational, and hyperfine structure of the NaK 33pi double minimum state and 43pi V-shaped potential. The double minimum and V-shaped character of these two potential wells arise from an avoided curve crossing. As a result, the NaK 33pi state exhibits strikingly different vibrational, rotational, fine, and hyperfine splittings in each region of the potential. In addition, high-lying ro-vibrational levels of the 3 3pi state, which are within 10 or 20 cm-1 of the nearest 43pi level with the same rotational quantum number J, and all levels of the 43pi state that lie below the NaK 33pi state dissociation limit experience significant non-adiabatic coupling effects and as a result are perturbed.;In the first step of the excitation process, a single-mode cw dye laser is used to excite 2(A)1Sigma+(v A, J) ∼ 1(b)3piO=0 (vb, J) mixed singlet-triplet "window" levels from thermally populated ro-vibrational ground state levels, 1(X)1Sigma+(v X, J+/-1). A single-mode cw Ti:Sapphire laser is then used to excite further the NaK molecules to various 33pi 0(vpi, Jpi) [43pi0( v'P,J 'P )] ro-vibrational levels, which are detected by observing direct 3 3pi0 → 1(a)3Sigma+ [43pi0 → 1(a)3Sigma +] fluorescence in the green spectral region.;From the 33piO=0 level energies, the inverse perturbation approximation method was used to construct a 3 3pi0 potential curve that reproduces the measured energies to ∼ 0.24 cm-1. One window level of the minor isotopomer 23Na41K was found and used to determine the absolute vibrational numbering of this state. In addition, two new 2(A)1Sigma +∼1(b)3piO=2 window levels were found and used to measure a few levels of the 33piO=2 and 43piO=2 states. The energy separations between ro-vibrational level O = 0 and O = 2 fine structure components were used to determine the NaK 33pi spin-orbit interaction constant and preliminary values of the NaK 43pi spin-orbit interaction constant, Av, as a function of vibrational quantum number. These values, along with the hyperfine splittings of the 3 3piO=2 and 43piO=2 level energies, were used in turn to obtain the Fermi contact constant, bF, as a function of v for each state.