  The potential of the $BC_1$ quantum elliptic model is a superposition of two
Weierstrass functions with doubling of both periods (two coupling constants).
The $BC_1$ elliptic model degenerates to $A_1$ elliptic model characterized by
the Lam\'e Hamiltonian. It is shown that in the space of $BC_1$ elliptic
invariant, the potential becomes a rational function, while the flat space
metric becomes a polynomial. The model possesses the hidden $sl(2)$ algebra for
arbitrary coupling constants: it is equivalent to $sl(2)$-quantum top in three
different magnetic fields. It is shown that there exist three one-parametric
families of coupling constants for which a finite number of polynomial
eigenfunctions (up to a factor) occur.
