We consider an ordinary differential equation of the fourth order, which is the first analogy to the first Painlev'e equation. By methods of Power Geometry, we find all power expansions of its solution near points z=0 and z=∞. For expansions of solutions near z=∞, we calculate exponential additions of the first, second and third levels. Our results confirm the conjecture that the equation determines new transcendetal functions. We also describe an algorithm of computation of a basis of a minimal lattice, containing a given finite set.

Mathematical problems and theory of numerical methods