We know that MOM estimate may not be unique. The most common example is Poisson distribution.
From my lecture notes, it said if we only consider $m_1 = \mu_1'$, then we have $\hat{\lambda} = \bar{X}$. While if we consider $m_1 = \mu_1'$ and $m_2 = \mu_2'$ together, then $m_2 = \lambda + m_1^2 =\lambda + \bar{X}^2$, which implies $\hat{\lambda} = m_2 - m_1^2$.
My question is why I cannot write $m_2 = \lambda + \lambda^2 $, so this will be a new estimate.
You can use a least squares approach, OLS (ordinary least squares) or GLS (generalized least squares), often used in the GMM (Generalized Method of Moments), as you have 2 conditions to find 1 estimator ($\hat \lambda$).
Using a simple OLS approach you will find the estimator solving: $$\hat \lambda=\underset{\lambda}{\mathrm{argmin}}\ (\lambda - m_1)^2+(\lambda-m_2+m_1^2)^2$$ Solving the f.o.c. for this convex optimization problem leads you to: $$\hat \lambda=\dfrac{m_1+(m_2-m_1^2)}{2}$$ just use the average of both MOM estimators. Look for GLS and GMM for details on more elaborated approaches.
