I will present two preprints [arXiv:2503.17816] and [arXiv:2503.19415], devoted to a 
basic equation of the form u''(x) + h(x) u(x) = 0. I will start with a discussion about 
generality of this equation and then introduce a family of metrics on an upper half plane, 
defined in terms of the function h(x). The sectional curvature of all these metrics is 
equal to -1, thus, they locally describe the same hyperbolic geometry. As the main result 
I will present a general solution of the equation in question expressed in terms 
of geodesic curves in the proposed two dimensional hyperbolic model. I will also show 
that an arbitrary pair of linearly independent solutions gives a diffeomorphism between 
the introduced geometry and Poincare upper half plane. In the second part I will discuss 
an equivalent approach in which the Riemannian geometry is replaced by an analogous 
Lorentzian geometry. Interestingly, it turns out that solutions of the associated Ricatti 
equation are at the same time geodesics in this second model. In the last part I will 
resort to complex Riemannian geometry with a holomorphic metric, in which I generalize 
previous result to the complex case with a holomorphic function h(z). Interestingly, 
this last geometry is locally diffeomorphic to a complex sphere with imaginary radius.