  We present a non-variational, kinetic energy operator approach to the
solution of quantum three-body problem with Coulomb interactions, based on the
utilization of symmetries intrinsic to the kinetic energy operator, i.e., the
three-body Laplacian operator with the respective masses. Through a four-step
reduction process, the nine dimensional problem is reduced to a one dimensional
coupled system of ordinary differential equations, amenable to accurate
numerical solution as an infinite-dimensional algebraic eigenvalue problem. A
key observation in this reduction process is that in the functional subspace of
the kinetic energy operator where all the rotational degrees of freedom have
been projected out, there is an intrinsic symmetry which can be made explicit
through the introduction of Jacobi-spherical coordinates. A numerical scheme is
presented whereby the Coulomb matrix elements are calculated to a high degree
of accuracy with minimal effort, and the truncation of the linear equations is
carried out through a systematic procedure
