We present a constructive method to convert the (two-dimensional) Jacobi--Perron evolution of $(x_1,x_2) \in [0,1]^2$ into the corresponding evolution of Podsypanin, and vice versa. A similar approach allows us to extend the result to Brun's algorithm. The method, based on the techniques of singularization and insertion, is built up in steps. From experiments, we assume that each step terminates after finitely many states almost everywhere.
"On the Singularization of the Two-Dimensional Jacobi--Perron Algorithm." Experiment. Math. 16 (4) 441 - 454, 2007.