First, you can find an irrational parametrization of the (n-2)-dimensional intersection surface as follows. Projecting along the first coordinate axis to the hyperplane x1=0, you get an ellipsoid in $R^{n-1}$. Now parametrize the new ellipsoid and invert the projection map; you will get a parametrization of the initial surface involving square roots [1].

Second, you may ask of a rational parametrization, i.e., parametrization by rational functions. Here is the answer in particular cases:

n=3. The curve you consider is called a cyclic [2]. It has a rational parametrization with complex coefficients only if it is singular. So, for general values of $\lambda_i$ there is no rational parametrization.

n=4. The surface you consider is called (an inverse stereographic image of) a cyclide [2,1] (not to be confused with a particular case of Dupin cyclide). Generically it is a del Pezzo surface of degree 4 from the point of view of complex algebraic geometry. Thus it always has a rational parametrization with complex coefficients, and even a biquadratic one [4]. But not always real coefficients will work because there are cyclides which are irreducible and disconnected simultaneously. As far as I know, no simple parametrization algorithm is available.

More information about algorithms for rational parametrization can be found in [3].

[1] H. Pottmann, L. Shi, M. Skopenkov, Darboux cyclides and webs from circles, submitted. [2] J. Coolidge, A treatise on the circle and the sphere, AMS Chelsea Publ., 1971. [3] J. Schicho, Rational Parametrization of Surfaces, J. Symbolic Computation (1998) 26, 1-29. [4] J. Schicho, The multiple conical surfaces, Contributions to Algebra and Geometry, Volume 42 (2001), No. 1, 71-87.

First, you can find an irrational parametrization of the (n-2)-dimensional intersection surface as follows. Projecting along the first coordinate axis to the hyperplane x1=0, you get an ellipsoid in $R^{n-1}$. Now parametrize the new ellipsoid and invert the projection map; you will get a parametrization of the initial surface involving square roots [1].

Second, you may ask of a rational parametrization, i.e., parametrization by rational functions. Here is the answer in particular cases:

n=3. The curve you consider is called a cyclic [2]. It has a rational parametrization with complex coefficients only if it is singular. So, for general values of $\lambda_i$ there is no rational parametrization.

n=4. The surface you consider is called (an inverse stereographic image of) a cyclide [2,1] (not to be confused with a particular case of Dupin cyclide). Generically it is a del Pezzo surface of degree 4 from the point of view of complex algebraic geometry. Thus it always has a rational parametrization with complex coefficients, and even a biquadratic one [4]. But not always real coefficients will work because there are cyclides which are irreducible and disconnected simultaneously. As far as I know, no simple parametrization algorithm is available.

More information about algorithms for rational parametrization can be found in [3].

[1] H. Pottmann, L. Shi, M. Skopenkov, Darboux cyclides and webs from circles, Computer Aided Geom. Design 29:1 (2012), 77-97.
[2] J. Coolidge, A treatise on the circle and the sphere, AMS Chelsea Publ., 1971. 
[3] J. Schicho, Rational Parametrization of Surfaces, J. Symbolic Computation (1998) 26, 1-29. 
[4] J. Schicho, The multiple conical surfaces, Contributions to Algebra and Geometry, Volume 42 (2001), No. 1, 71-87.

First, you can find an irrational parametrization of the (n-2)-dimensional intersection surface as follows. Projecting along the first coordinate axis to the hyperplane x1=0, you get an ellipsoid in $R^{n-1}$. Now parametrize the new ellipsoid and invert the projection map; you will get a parametrization of the initial surface involving square roots [1].

Second, you may ask of a rational parametrization, i.e., parametrization by rational functions. Here is the answer in particular cases:

n=3. The curve you consider is called a cyclic [2]. It has a rational parametrization with complex coefficients only if it is singular. So, for general values of $\lambda_i$ there is no rational parametrization.

n=4. The surface you consider is called (an inverse stereographic image of) a cyclide [2,3] (not to be confused with a particular case of Dupin cyclide). Generically it is a del Pezzo surface of degree 4 from the point of view of complex algebraic geometry. Thus it always has a rational parametrization with complex coefficients, and even a biquadratic one [4]. But not always real coefficients will work because there are cyclides which are irreducible and disconnected simultaneously. As far as I know, no simple parametrization algorithm is available.

More information about algorithms for rational parametrization can be found in [3].

[1] H. Pottmann, L. Shi, M. Skopenkov, Darboux cyclides and webs from circles, submitted. [2] J. Coolidge, A treatise on the circle and the sphere, AMS Chelsea Publ., 1971. [3] J. Schicho, Rational Parametrization of Surfaces, J. Symbolic Computation (1998) 26, 1-29. [4] J. Schicho, The multiple conical surfaces, Contributions to Algebra and Geometry, Volume 42 (2001), No. 1, 71-87.

First, you can find an irrational parametrization of the (n-2)-dimensional intersection surface as follows. Projecting along the first coordinate axis to the hyperplane x1=0, you get an ellipsoid in $R^{n-1}$. Now parametrize the new ellipsoid and invert the projection map; you will get a parametrization of the initial surface involving square roots [1].

Second, you may ask of a rational parametrization, i.e., parametrization by rational functions. Here is the answer in particular cases:

n=3. The curve you consider is called a cyclic [2]. It has a rational parametrization with complex coefficients only if it is singular. So, for general values of $\lambda_i$ there is no rational parametrization.

n=4. The surface you consider is called (an inverse stereographic image of) a cyclide [2,1] (not to be confused with a particular case of Dupin cyclide). Generically it is a del Pezzo surface of degree 4 from the point of view of complex algebraic geometry. Thus it always has a rational parametrization with complex coefficients, and even a biquadratic one [4]. But not always real coefficients will work because there are cyclides which are irreducible and disconnected simultaneously. As far as I know, no simple parametrization algorithm is available.

More information about algorithms for rational parametrization can be found in [3].

[1] H. Pottmann, L. Shi, M. Skopenkov, Darboux cyclides and webs from circles, submitted. [2] J. Coolidge, A treatise on the circle and the sphere, AMS Chelsea Publ., 1971. [3] J. Schicho, Rational Parametrization of Surfaces, J. Symbolic Computation (1998) 26, 1-29. [4] J. Schicho, The multiple conical surfaces, Contributions to Algebra and Geometry, Volume 42 (2001), No. 1, 71-87.