Uniform (m)-condition and Strong Milnor fibrations

In this paper we study the Milnor fibrations associated to real analytic map germs $\psi:(\mathbb{R}^{m},0) \to (\mathbb{R}^2,0)$ with isolated critical point at $0\in \mathbb{R}^{m}$. The main result relates the existence of called Strong Milnor fibrations with a transversality condition of a convenient family of analytic varieties with isolated critical points at the origin $0\in \mathbb{R}^{m}$, obtained by projecting the map germ $\psi$ in the family $L_{-\theta}$ of all lines through the origin in the plane $\mathbb R^{2}.$