Modified Runge-Kutta methods are well suited for fulfilling properties that require coefficients depending on step-length. By a simple perturbation of very few coefficients we may produce various function-fitted methods and avoid the overhead of evaluating all of them in every step. In this paper we present the formula of determining the extra algebraic equations of condition generated by the major subcategory of these methods. Also phase-lag and phase-fitted properties are analyzed for this case. Finally a specific phase-fitted pair of orders 6(5) is given.