  We present in total fifteen potentials for which the stationary Klein-Gordon
equation is solvable in terms of the confluent Heun functions. Because of the
symmetry of the confluent Heun equation with respect to the transposition of
its regular singularities, only nine of the potentials are independent. Four of
these independent potentials are five-parametric. One of them possesses a
four-parametric ordinary hypergeometric sub-potential, another one possesses a
four-parametric confluent hypergeometric sub-potential, and one potential
possesses four-parametric sub-potentials of both hypergeometric types. The
fourth five-parametric potential has a three-parametric confluent
hypergeometric sub-potential, which is, however, only conditionally integrable.
The remaining five independent Heun potentials are four-parametric and have
solutions only in terms of irreducible confluent Heun functions.
