True macro lenses come in fixed focal length. Are there any macro lensens which have variable focal lengths? And please explain why yes or why not?
It depends on how you define fixed and variable. As you change the focus distance of many prime lenses, including some macro lenses, the actual focal length changes a little bit. Most fixed focal length lens' focal lengths are defined when the lens is focused on infinity and the light focused at the film/sensor plane is collimated when entering the lens. With Macro lenses that are also capable of focusing collimated light at the sensor plane the difference in focal length when focused at the minimum focus distance (MFD) will be greater than with a more typical lens that can't focus as close. A simple single element lens must be twice as far from the camera's imaging plane as its focal length to provide 1:1 magnification of an object in focus at the same distance as the camera's imaging plane but on the other side of the lens.
It also depends on how you define True macro lenses. If you are using a lens with a maximum magnification (MM) of 0.5x on a camera with a 2x crop factor sensor a resulting 8x10 print will show the subject the same size as if you had used a lens with a MM of 1.0x and a full frame camera.
The Canon MP-E 65mm 1-5x Macro lens is listed as a fixed focal length lens, but for all practical purposes the focal length is meaningless. At 1x the angle of view (AoV) is about what one would expect for a 65mm simple single element lens focused at unity (or 1:1 magnification with the subject 260mm from the camera's imaging plane) which is about the same angle of view as a 130mm lens focused at infinity. At 5x the AoV is 1/5 that, or what one would expect from an approximately 325mm lens focused at unity, which gives an AoV about equal to a 650mm lens focused at infinity. The lens can only focus at a single, fixed specific distance at any particular magnification setting. At 1x it has about 100mm working distance (the distance from the front of the lens to the point of focus). By 5x the working distance is only 41mm. Since the lens can't focus collimated light onto the sensor when connected to a camera with the registration distance for which it was designed at any setting, there is no real way to express focal length in the conventional sense.
It also depends on how you define zoom lens. Another clue that the MP-E 65mm is a unique kind of (sort of) zoom lens is the chart included on page 8 of the MP-E 65mm 1-5x Macro User Manual. As the magnification is increased, the effective f-number for any given aperture setting also increases as one would expect when the same sized opening of the diaphragm is used for a longer focal length lens.
Nikon makes a 70-180mm macro zoom lens. It focuses down to a 1:1.3 magnification ratio -- not quite what is considered "true macro" -- but with the 6T close-up lens it gets to 1:1. Supposedly quite good, though I have no first-hand experience.
I would argue that most modern macro lenses, specifically the ones with internal (rear) focussing, do NOT keep a constant focal length. If f is constant then for 1:1 the working distance is 2f, and from object to sensor plane is 4f plus the thickness of the lens. The spec of most lenses at 1:1 reads far less than that, ergo f becomes much shorter than what is written on the lens.
If you buy a 100 mm lens then you expect a working (object) distance of 200 mm at 1:1, but you get less, perhaps only 150 or 120 mm. So in reality you have a 75 or 60 mm lens when it matters. Duh.
I have an old 100 mm Cosina macro lens (unit focussing) that goes down to 2:1, i.e. 300 mm object distance and 150 mm to the sensor. For reaching 1:1 you have to screw on a +3.3 diopter that makes it a 75 mm lens, i.e. 150 mm object distance and the same 150 mm to the sensor. This makes sense: to get more magnification you need a stronger lens, a shorter f. But for that I lose 50 mm of working distance. I can also use macro rings, e.g. +50 mm gives me my 1:1 at 200 mm working distance, but now I lose 2 stops of light (instead of 1 ?).
So modern macro lenses "cheat" bij reducing the focal length (= zooming OUT) when getting closer. This gives you more magnification and more light but less working distance than the focal length at infinity number suggests. It is a useless number for the purpose of macro photography. Give us the real numbers !
PS there is no zoom control, it is done automagically by the internal focussing.