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Peel, deseed and cut the apples into eighths. Pour the lemon juice over the apples and sprinkle with a little caster sugar so that they do not darken.
Heat the oven to 190C/fan 170C/gas 5. Beat the eggs with the remaining caster sugar in a mixer until you have a thick, pale, mousse-like mixture. Sift over the flour and baking powder, and fold it in followed by the soured cream, rum and butter.
Butter the inside of a 23cm springform tin and arrange the apples on the base. Spoon the batter on top and level it with a spatula. Put the cake in the oven and bake for 50 minutes-1 hour without opening the door. Cool the cake and then turn it out onto a plate, and sprinkle it with icing sugar to finish.