On solvability of equations with endomorphisms in nilpotent groups

We prove that the conjugacy, twisted conjugacy and bi-twisted conjugacy problems, and the corresponding search problems, are decidable for the class $\mathbf{N}_{fg} $ of all finitely generated nilpotent groups. Also we give a finite description of the equalizer of any pair of endomorphisms of arbitrary group in the class $\mathbf{N}_{fg}$.