Supplemental code which solves magnetic breakdown problem near generalized van Hove singularities; spectral equations for several example lattice models; Kwant simulations for tight-binding models of these lattice models in magnetic field. Structure of the code files: 1. all_functions_mod.py - supplemental file with functions that improve matplotlib plots. 2. analytical_energy_levels_square_lat.py, analytical_energy_levels_a3_square_lat.py, analytical_energy_levels_triang_lat.py - files that contain implementation of analytic expressions (solution of spectral equation and dynamical phases calculation) for the square lattice with usual and A3 van Hove singularities as well as for triangular lattice with Monkey saddles. 3. landau_levels.py - code that is a part of Kwant library, performs discretization of symbolic Hamiltonian into oscillator basis. 4. module.py - module that contains main functions related to angle separation of the modes and proper S-matrix calculation. 5. kwant_systems_leads_and_finite_slabs.py - this file contains all definitions of kwant systems for tight-binding simulations. Finite lattices for conductance as well as infinite leads for spectrum. 6. scattering_phases.py, tunneling_amplitudes.py - code that separates scattering phases and tunneling amplitudes used for substition into spectral equations. 7. Schematic_figures_for_manuscript_v3.ipynb - jupyter notebook with code that generates schematic figures for mapping and S-matrix results. 8. conductance_vs_spectrum_plotting.ipynb - jupyter notebook that contains code for spectrum comparison and conductance calculations as well as generation of corresponding plots. Conductance data is also stored in this notebook. Calculations for p=100 are presented, but not used for the plots in the manuscript. Data folder contains: 1. A3_point_N20004_t1_(value of t1).npy - files with scattering matrix of A3 points for different values of regularizing parameter called t1 here (0.01, 0.005 etc) 2. l_levels_square_p='+str(p)+'_en_range='+'top'+'.npy', q1_set_square_p='+str(p)+'_en_range='+'top.npy', q2_set_square_p='+str(p)+'_en_range='+'top.npy' - files that contain data for landau level energies, wave number set along x direction amd wave number set along y direction for different values of magnetic flux (defined by p=40, 100) for square lattice. 3. Same files as in 2 for triangular and square lattice with A3 point are presented. 4. s_matrix_monkey_saddle.npz, s_matrix_a3_saddle.npz, phase_scatt_monkey_saddle.npz - data files with scattering matrices for Monkey and A3 saddle points, and separately data for scattering phase for Monkey saddle. Inserted into analytical spectral equation.