In a sense, there is only one fixed point. Look to the definition (http://www.bipm.org/en/CGPM/db/13/4):
The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
There's a lot hiding in that simple definition. One key concept is that of thermodynamic temperature. A temperature scale that does not have zero at absolute zero is not a thermodynamic temperature scale. So in this sense, the Kelvin scale has two fixed points, absolute zero and the triple point of water.
Another issue hiding in that definition is that it uses the triple point of water. Purified water from an East African lake, from the oceans, or Antarctic ice have slightly different triple points, slightly different freezing points, and slightly different boiling points. The isotopic composition of water varies with latitude. To get around this issue, the water used in determining the triple point of water must have a very specific isotopic composition specified by the Vienna Standard Mean Ocean Water (VSMOW) standard.
But , then another section in the same book says :
On the Kelvin Scale, the lower fixed point is taken as 273.15 K and the upper fixed point as 373.15 K.
This portion of your text is extremely outdated. The triple point of water has been used as the sole fixed point in the SI temperature scale since 1954. Using the freezing and boiling point of water as defining temperature is a discarded concept. Textbooks are oftentimes a decade or so out-of-date; that happens all the time. On the other hand, a textbook that is over six decades out-of-date is unacceptable.
Even using the triple point of water as a fixed point will most likely soon be discarded, at least in theory. If all goes according to plan, the only fixed point will be absolute zero. Temperature in the International System of Units will soon be defined by making the Boltzmann constant a defined value.
In practice, measuring temperature is rather difficult. The International Temperature Scale, which attempts to be a practical realization of the SI concept of the kelvin, has 14 different fixed points (excluding absolute zero). The melting and boiling points of water are not amongst those fixed points. The triple point of water (VSMOW) is.