  We show that the Natanzon family of potentials is necessarily dropped into a
restricted set of distinct potentials involving a fewer number of independent
parameters if the potential term in the Schr\"odinger equation is proportional
to an energy-independent parameter and if the potential shape is independent of
both energy and that parameter. In the hypergeometric case only six such
potentials exist, all five-parametric. Among these, only two (Eckart,
P\"oschl-Teller) are independent in the sense that each cannot be derived from
the other by specifications of the involved parameters. Discussing the
solvability of the Schr\"odinger equation in terms of the single-confluent Heun
functions, we show that in this case there exist in total fifteen
seven-parametric potentials, of which independent are nine. Six of the
inde-pendent potentials present different generalizations of the hypergeometric
or confluent hypergeometric ones, while three others do not possess
hypergeometric sub-potentials. The result for the double- and bi-confluent Heun
equations produces the three independent double- and five independent
bi-confluent six-parametric Lamieux-Bose potentials, and the general
five-parametric quartic oscillator potential for the tri-confluent Heun
equation.
