This is a problem I found trying to find some properties related to exchangeable sequences. Anyway, I am not able to find a characterization of all solutions.. I know there are at least two completely different solutions, is it possible to find all of them?
Problem: Find all pairs of (finite non-negative) measures $\alpha, \beta$ defined on the Borel $\sigma$-algebra of $[0,1]$ satisfying the following three conditions:
(i) $\int_0^1\alpha(\mathrm{d}x)=2+\int_0^1\beta(\mathrm{d}x)$
(ii) $\int_0^1 x\alpha(\mathrm{d}x)=1+\int_0^1x\beta(\mathrm{d}x)$
(iii) $\int_0^1 x^2\alpha(\mathrm{d}x)=\int_0^1x^2\beta(\mathrm{d}x)$
The difference $\gamma = \alpha - \beta$ is a signed measure. The corresponding bounded linear functional $\phi$ on $C[0,1]$ must satisfy $\phi(1) = 2$, $\phi(x) = 1$, $\phi(x^2) = 0$. Since $1$, $x$ and $x^2$ are linearly independent, this defines an affine subset of $M[0,1]$ of codimension $3$. A typical $3$-dimensional subspace of $M[0,1]$ will intersect it.
For example, consider the measures $\delta_0$ (the unit mass at $0$), $\delta_1$ (unit mass at $1$) and $m$ (Lebesgue measure). The linear combination $\gamma = c_1 \delta_0 + c_2 \delta_1 + c_3 \delta_3$ satisfies the conditions if $$\eqalign{c_1 + c_2 + c_3 &= 2\cr c_2 + c_3/2 &= 1\cr c_2 + c_3/3 &= 0\cr}$$ which has solution $c_1 = -2$, $c_2 = -2$, $c_3 = 6$. Let's write this as $\gamma_0 = -2 \delta_0 - 2 \delta_1 + 6 m$.
The general signed measure $\gamma$ on $[0,1]$ satisfying $\int 1\; d\mu = 2, \int x \; d\mu = 1, \int x^2 \; d\mu = 0$ is $\gamma = \gamma_0 + \nu$ where $\nu$ is a signed measure with $\int 1\; d\nu = \int x \; d\nu = \int x^2\; d\nu = 0$. For any such $\gamma$, let $\gamma = \gamma_+ - \gamma_-$ be its Hahn decomposition into positive and negative parts, and let $\rho$ be any finite positive Borel measure. Then $\alpha = \gamma_+ + \rho$, $\beta = \gamma_- + \rho$ is a solution to your problem.