  This paper uses Lie symmetry methods to analyze boundary crossing
probabilities for a large class of diffusion processes. We show that if
Fokker--Planck--Kolmogorov equation has non-trivial Lie symmetry, then boundary
crossing identity exists and depends only on parameters of process and
symmetry. For time-homogeneous diffusion processes we found necessary and
sufficient conditions of symmetries' existence. This paper shows that if drift
function satisfy one of a family of Ricatti equations, then the problem has
nontrivial Lie symmetries. For each case we present symmetries in explicit
form. Based on obtained results, we derive two-parametric boundary crossing
identities and prove its uniqueness. Further, we present boundary crossing
identities between different process. We show, that if the problem has 6 or 4
group of symmetries then the first passage time density to any boundary can be
explicitly represented in terms of the first passage time by Brownian motion or
Bessel process. Many examples are presented to illustrate the method.
