High resolution spectra of the 1 (1)Pi(v'=13-69,J')<-- X (1)Sigma(+)(v",J") and 2 (1)Pi(v'=0-13,J')<-- X (1)Sigma(+)(v',J') transitions of the (KRb)-K-39-Rb-85 molecule have been measured with the technique of Doppler-free optical-optical double resonance polarization spectroscopy (OODRPS). Molecular constants of the 1 (1)Pi(v=13-69) and 2 (1)Pi(v=0-13) levels have been determined, and potential energy curves constructed by the RKR method. The RKR potential of the 1 (1)Pi state was found to have a distortion at outer wall, which originates from an avoided crossing of two (1)Pi states. The perturbations between the 1 (1)Pi(v(1),J) and 2 (1)Pi(v(2),J) levels were found from the energy shifts of the rotational levels. The magnitude of the nonadiabatic interaction between the 1 (1)Pi(v(1)=54) and 2 (1)Pi(v(2)=9) levels, < 1 (1)Pi(v(1)=54)\T-N\2 (1)Pi(v(2)=9)>, was evaluated to be 2.2 cm(-1) by a least squares fitting to the energy shifts of the 1 (1)Pi(v(1)=54,J=20-33) levels. The line intensities were observed to change dramatically around the maximum energy shift. These intensity anomalies are interpreted as an interference effect, which occurs when two interacting levels have comparable transition moments. A remarkable line broadening was observed for the transitions to the 1 (1)Pi(v greater than or equal to 63) levels, and it was identified as originating from the predissociation to K(4s(2)S(1/2))+Rb(5p(2)P(1/2)) atoms. The dissociation energies of the X (1)Sigma(+), 1 (1)Pi, and 2 (1)Pi states have been determined to be 4217.4 +/- 0.8, 2021.5 +/- 0.8, and 1050.0 +/-0.8 cm(-1), respectively.