This work presents high-resolution emission spectra measurements of the Herzberg band system, which has not been observed and analysed in the 13C17O isotopologue so far. Bands C → A (0,1), (0,2) and (0,3) were recorded in a region at 22950-26050 cm-1 using high-accuracy dispersive optical spectroscopy. The 13C17O molecules were formed and excited in a stainless steel hollow-cathode lamp with two anodes. All 224 rovibrational spectra lines, up to Jmax = 30, were precisely measured with an accuracy of about 0.0030 cm-1 and rotationally analysed. In this work the following have been determined in 13C17O for the first time: the merged rotational constants of the C1Σ+(ν = 0) Rydberg state and the individual rotational constants of the A1Π(ν = 3) state, as well as the rotational and vibrational equilibrium constants for the C1Σ+ state, the band origins of the C → A system, the isotope shifts, and the ΔGC1/2 vibrational quantum. The combined analysis of the Herzberg bands obtained now and the Ångström (B1Σ+ → A1Π) system analysed earlier (R. Hakalla et al., J. Phys. Chem. A, 2013, 117, 12299 and R. Hakalla et al., J. Mol. Spectrosc., 2012, 272, 11) yielded a precisely relative characteristic of the C1Σ+(ν = 0) and B1Σ+(ν = 0 and 1) Rydberg states in the 13C17O molecule, among others νCB00, νCB01 vibrational quanta. Also, many molecular constant values of the C1Σ+ state in the 12C16O, 12C17O, 13C16O, 12C18O, and 13C18O isotopologues were determined, which have not been published so far, as well as the RKR turning points, Franck-Condon factors, relative intensities, r-centroids for the Herzberg band system and the main, isotopically invariant parameters of the C1Σ+ state in the CO molecule within the Born-Oppenheimer approximation. In the A1Π(ν = 3) state of the 13C17O molecule, extensive, multi-state rotational perturbations were found, which were analysed and substantiated in detail. The vibrational level ν = 0 of the C1Σ+ state was analysed, paying special attention to possible irregularities, and no noticeable perturbations were found in it up to the observed Jmax.