  The two-time Green function method is employed to explore the effect of the
biquadratic exchange interaction on the phase diagram of a $d$-dimensional
spin-1 transverse XY model with single-ion anisotropy close to the
magnetic-field-induced quantum critical point. We work at level of the
Anderson-Callen-decoupling-like framework for both easy-plane and easy-axis
single-ion anisotropy. The structure of the phase diagram is analyzed with
analytical estimates and numerical calculations by adopting Tyablikov-like
decouplings for the exchange higher order Green functions in the equation of
motion. For dominant bilinear short-range exchange interaction the structure of
the phase diagram close to the quantum critical point remains qualitatively the
same of that in absence of biquadratic interaction including reentrant critical
lines. When the biquadratic exchange becomes increasingly dominant its role
appears more effective and tends to reduce or destroy the reentrant character
of the critical lines.
