Industrial gearboxes are used in machines and mechanisms across many different industries. The drive of a quarter-turn ball valve manufactured based on worm or spiroid gears has a rather low efficiency and a high starting torque which often prevents pipelines from fail-safe opening or closing under harsh operating conditions. One way to improve petroleum equipment drives that provide its serviceability under the abovementioned conditions is to use bevel precessional gears with a small shaft angle. To date, bevel gears with a small shaft angle with double-concave-convex teeth have been studied. The paper reviews a new way of obtaining the correction values of machine settings for tooth surface modification achieved by crowning the pinion and gear in the longitudinal direction of a straight bevel gear with a small shaft angle with a non-generated gear and a generated pinion. It is applied to kinematically simple machines. In order to exclude the edge contact of gears and to decrease the sensitivity of the pinion and gear to errors in tooth surfaces and to the relative positions of the mating gears, the authors proposed to manufacture longitudinally crowned gear tooth surfaces, and based on the theory of gearing the authors obtained relationships describing the gear tooth surface. A program was developed to determine the position of the lines of action in meshing in a gear with the localized contact. The authors studied a change in the position of the lines of action if there are position errors of the mating pinion and gear.