Remove the zest from the lemons. Mix them with the sugar in a large bowl and let stand 10 min. Preheat the oven at 320°F (160°C).
Melt the butter in a bowl and let it cool.
In the lemon sugar mixture, add the eggs one by one, stirring between each one.
Add the cream, rum and salt and mix.
In a bowl, mix the flour and baking powder. Add them to the previous preparation in 3 times, stirring each time.
Finally add the butter
Put the cake batter in a large cake pan and bake for 70 to 80 minutes in the preheated oven at 320°F (160°C).
Check that the cake is cooked by inserting a knife which should come out dry.
Prepare the syrup: heat the sugar and water in a saucepan. Remove from the heat at the first broth. Add the lemon juice off the heat.
At the exit of the oven, pour the syrup (warm or cold) on the cake (still in its mould). Leave to cool for at least 1 hour before unmoulding.
Good to know: with these proportions, that's one big cake and two mini ones.