Variational equations in parametric variables and transformation of their solutions

An analytical relationship is derived which connects the matrix of partial derivatives of the current state vector with respect to the initial conditions with the matrix of solutions to variational equations in parametric variables. These equations correspond to the equations of motion obtained using the generalized coordinate-time transformation admitting an increase of the number of unknown parameters. Particular cases of transformations leading to the variational equations in regularizing variables of Sperling-Burdet and Kustaancheimo-Stiefel are considered. All formulas necessary in these cases for determination of the matrix of isochronous derivatives are obtained.