  We develop a minimal non-BCS model for the CuO$_2$ planes with the on-site
Hilbert space reduced to only three effective valence centers CuO$_4$ with
different charge, conventional spin, and orbital symmetry, combined in a charge
triplet. Using the S=1 pseudospin algebra we introduce an effective
spin-pseudospin Hamiltonian. To illustrate the possibilities of the molecular
field approximation we start with the analysis of the atomic and the "large
negative-$U$" limits of the model in comparison with the Bethe cluster
approximation, classical and quantum Monte Carlo methods. Both limiting systems
exhibit the phase separation effect typical of systems with competing order
parameters. The $T$\,-\,$n$ phase diagrams of the complete spin-pseudospin
model were reproduced by means of a site-dependent variational approach within
effective field approximation typical for spin-magnetic systems. Limiting
ourselves to two-sublattice approximation and $nn$-couplings we arrived at
several N\'{e}el-like phases in CuO$_2$ planes for parent and doped systems
with a single nonzero local order parameter: antiferromagnetic insulator,
charge order, glueless $d$-wave Bose superfluid phase, and unusual metallic
phase. However, the global minimum of free energy is realized for phase
separated states which are bounded by the third-order phase transition line
$T^{\star}(n)$, which is believed to be responsible for the onset of the
pseudogap phenomenon. With a certain choice of the Hamiltonian parameters the
model phase diagrams can quite reasonably reproduce the main features of
experimental phase diagrams for T- and T$^{\prime}$-cuprates and novel
nickelates. The superconducting phase of cuprates/nickelates is determined by
the on-site composite boson transport, it is not a consequence of pairing of
doped holes/electrons, but represents one of the possible phase states of
parent systems.
