Is it permissible for the one paying Zakat (the muzakki) to give the Zakat of his wealth to his poor mother, especially since she has no one to support her but him?

It is obligatory for a child, whether male or female, to financially support their poor parents if the child is wealthy, provided that what they spend on them is surplus to their own needs and the needs of their spouse and dependents. It is also obligatory for one who is able to earn a living to work in order to support their parents if necessary. This obligatory expenditure cannot be counted as zakat (definition) funds, because it fulfills your obligation of support and its benefit returns to you. Filial piety towards parents and showing them kindness brings blessings and abundant provision.