The book first expounds the systematic theory of solutions in integers of algebraic indefinite equations from two unknowns. Based on a single method that synthesizes the ideas of the theory of algebraic and transcendental numbers, all the main types of equations are analyzed. The main attention is paid to the study of the influence of equation parameters on the values ​ ​ of their solutions. All main and some auxiliary results are obtained only a few years ago, many are published for the first time.