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With a variation the sun is directly over the equator all the time. That's why it is hottest there. Time on Earth is a latitude thing not a matter of longitude.

However, to answer "when is the Sun directly over the equator as it transition from Northern to Southern hemispheric dominance and back?" is at the Autumnal and Vernal Equinox. At this time, no matter when you are on the planet, day and night are both 12 hours. After that event, the day will grow longer or shorter until the summer or wither Solstice.
The last was 22 SEP 2009 @ 22:18 (Autumnal, or September) and the next will be 20 MAR 2010 @ 17:32 (Vernal, or March)
It is growing more common for them to be revered to by the months in which they occur.

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