We show that a real analytic family of real analytic functions admits a modified analytic trivialization (MAT) via an appropriate toric modification, if their weighted initial forms define isolated singularities near zero.
Toshizumi FUKUI. Laurentiu PAUNESCU. "Modified analytic trivialization for weighted homogeneous function-germs." J. Math. Soc. Japan 52 (2) 433 - 446, April, 2000. https://doi.org/10.2969/jmsj/05220433