  We start by introducing a nonlinear involution operator which maps the space
of solutions of Sturm-Liouville equations into the space of solutions of the
associated equations which turn out to be nonlinear ordinary differential
equations. We study some algebraic and analytical properties of this involution
operator as well as some properties of a two-parameter family of operators
describing the set of solutions of Sturm-Liouville equations. Next, we show how
a specific composition of these mappings allows to connect, by means of a
simple analytical expression, the law of the first passage time of a Brownian
motion over a curve to a two-parameter family of curves. We offer three
different proofs of this fact which may be of independent interests. In
particular, one is based on the construction of parametric time-space harmonic
transforms of the law of some Gauss-Markov processes. Another one, which is of
algebraic nature, relies on the Lie group symmetry methods applied to the heat
equation and reveals that our two-parameter transformation is the unique
non-trivial one.
